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PHILIPPE RIGOLLET: OK, so the
course you're currently sitting

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in is 18.650.

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00:00:27,670 --> 00:00:29,570
And it's called
Fundamentals of Statistics.

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And until last spring, it
was still called Statistics

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for Applications.

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It turned out that really, based
on the content, "Fundamentals

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of Statistics" was a
more appropriate title.

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00:00:42,399 --> 00:00:43,940
I'll tell you a
little bit about what

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00:00:43,940 --> 00:00:46,231
we're going to be covering
in class, what this class is

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about, what it's not about.

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I realize there's
several offerings

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in statistics on campus.

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So I want to make sure that
you've chosen the right one.

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And I also understand
that for some of you,

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it's a matter of scheduling.

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I need to actually
throw out a disclaimer.

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I tend to speak too fast.

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I'm aware that.

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Someone in the back, just
do like that when you

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have no idea what I'm saying.

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Hopefully, I will repeat
myself many times.

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So if you average
over time, you'll

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see that statistics
will tell you

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that you will get the right
message that I was actually

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trying to stick to send.

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All right, so what are
the goals of this class?

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The first one is basically
to give you an introduction.

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No one here is expected to
have seen statistics before,

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but as you will see,
you are expected

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to have seen probability.

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And usually, you do
see some statistics

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in a probability course.

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So I'm sure some of
you have some ideas,

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but I won't expect anything.

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And we'll be using mathematics.

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Math class, so there's going
to be a bunch of equations--

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not so much real data
and statistical thinking.

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00:01:52,532 --> 00:01:54,740
We're going to try to provide
theoretical guarantees.

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We have two estimators
that are available for me--

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how theory guides me to choose
between the best of them,

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how certain can I be of my
guarantees or prediction?

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00:02:06,320 --> 00:02:08,549
It's one thing to
just bid out a number.

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00:02:08,549 --> 00:02:10,590
It's another thing to put
some error bars around.

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And we'll see how to build
error bars, for example.

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You will have your
own applications.

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I'm happy to answer questions
about specific applications.

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But rather than trying
to tailor applications

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to an entire institute,
I think we're

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going to work with pretty
standard applications,

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mostly not very serious ones.

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And hopefully, you'll be able
to take the main principles back

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with you and apply them to
your particular problem.

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What I'm hoping that you will
get out of this class is that

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when you have a real-life
situation-- and by "real life",

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I mean mostly at MIT, so some
people probably would not call

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that real life--

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their goal is to formulate
a statistical problem

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in mathematical terms.

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If I want to say,
is a drug effective,

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that's not in
mathematical terms,

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I have to find out
which measure I want

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to have to call it effective.

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Maybe it's over a
certain period of time.

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So there's a lot of things
that you actually need.

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And I'm not really
going to tell you

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how to go from the application
to the point you need to be.

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But I will certainly
describe to you

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what point you need to be at
if you want to start applying

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statistical methodology.

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Then once you understand
what kind of question

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you want to answer--

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do I want a yes/no answer,
do I want a number,

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do I want error bars, do
I want to make predictions

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five years into future, do
I have side information,

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or do I not have side
information, all those things--

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based on that,
hopefully, you will

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have a catalog of
statistical methods

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that you're going to be able to
use and apply it in the wild.

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And also, no statistical
method is perfect.

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Some of the math people have
agreed upon over the years,

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and people understand
that this is the standard.

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But I want you to be
able to understand

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what the limitations are,
and when you make conclusions

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based on data, that those
conclusions might be erroneous,

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for example.

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All right, more practically, my
goal here is to have you ready.

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So who has taken, for example,
a machine-learning class here?

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All right, so many of you,
actually-- maybe a third

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have taken a
machine-learning class.

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So statistics has somewhat
evolved into machine

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learning in recent years.

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And my goal is to
take you there.

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So machine learning has a
strong algorithmic component.

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So maybe some of you have
taken a machine-learning class

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that displays mostly the
algorithmic component.

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But there's also a
statistical component.

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The machine learns from data.

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So this is a statistical track.

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And there are some statistical
machine-learning classes

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that you can take here.

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They're offered at the
graduate level, I believe.

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But I want you to be ready to
be able to take those classes,

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having the statistical
fundamentals to understand

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what you're doing.

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And then you're going to be able
to expand to broader and more

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sophisticated methods.

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Lectures are here from 11:00 to
12:30 on Tuesday and Thursday.

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Victor-Emmanuel will also be--

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and you can call him Victor--

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will also be holding
mandatory recitation.

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So please go on Stellar
and pick your recitation.

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It's either 3:00 to 4:00 or
4:00 to 5:00 on Wednesdays.

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And it's going to be mostly
focused on problem-solving.

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They're mandatory in the sense
that we're allowed to do this,

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but they're not going to
cover entirely new material.

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But they might cover
some techniques

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that might save you some time
when it comes to the exam.

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So you might get by.

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Attendance is not going to be
taken or anything like this.

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But I highly
recommend that you go,

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because, well,
they're mandatory.

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So you cannot really complain
that something was taught only

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in recitation.

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So please register
on Stellar for which

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of the two recitations
you would like to be in.

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They're capped at 40, so
first come, first served.

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Homework will be due weekly.

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There's a total of
11 problem sets.

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I realize this is a lot.

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Hopefully, we'll
keep them light.

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I just want you to
not rush too much.

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The 10 best will
be kept, and this

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will count for a total of
30% of the final grade.

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There are due Mondays
at 8:00 PM on Stellar.

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And this is a new thing.

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We're not going to use the boxes
outside of the math department.

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We're going to use
only PDF files.

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Well, you're always welcome
to type them and practice

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your LaTeX or Word typing.

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I also understand that this
can be a bit of a strain,

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so just write them down
on a piece of paper,

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use your iPhone, and
take a picture of it.

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Dropbox has a nice, new--

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so try to find something
that puts a lot of contrast,

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especially if you use
pencil, because we're going

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to check if they're readable.

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And this is your responsibility
to have a readable file.

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I've had over the years--

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not at MIT, I must
admit-- but I've

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had students who actually
write the doc file

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and think that
converting it to a PDF

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consists in erasing
the extension doc

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and replacing it by PDF.

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This is not how it works.

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So I'm sure you
will figure it out.

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Please try to keep
them letter-sized.

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This is not a
strict requirement,

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but I don't want to
see thumbnails, either.

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You are allowed to have
two late homeworks.

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And by late, I
mean 24 hours late.

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No questions asked.

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You submit them,
this will be counted.

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You don't have to send
an email to warn us

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or anything like this.

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Beyond that, even that
you have one slack

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for one 0 grade and slack
for two late homeworks,

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you're going to have to come
up with a very good explanation

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why you need actually more
extensions than that, if you

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ever do.

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And particularly,
you're going to have

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to keep track about why you've
used your three options before.

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There's going to
be two midterms.

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One is October 3, and
one is November 7.

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They're both going to be
in class for the duration

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of the lecture.

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When I say they last for
an hour and 20 minutes,

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it does not mean that
if you arrive 10 minutes

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before the end of
lecture, you still

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get an hour and 20 minutes.

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It will end at the
end of lecture time.

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For this as well, no pressure.

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Only the best of the
two will be kept.

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And this grade will count
for 30% of the grade.

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This will be closed-books
and closed-notes.

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The purpose is for you to-- yes?

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AUDIENCE: How many midterms
did you say there are?

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PHILIPPE RIGOLLET: Two.

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AUDIENCE: You said the best
of the two will be kept?

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PHILIPPE RIGOLLET: I
said the best of the two

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will be kept, yes.

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AUDIENCE: So both the
midterms will be kept?

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PHILIPPE RIGOLLET: The best
of the two, not the best two.

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AUDIENCE: Oh.

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PHILIPPE RIGOLLET: We will add
them, multiply the number by 9,

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and that will be grade.

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No.

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I am trying to be nice, there's
just a limit to what I can do.

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All right, so the goal is
for you to learn things

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00:09:02,340 --> 00:09:04,020
and to be familiar with them.

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00:09:04,020 --> 00:09:05,700
In the final, you
will be allowed

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00:09:05,700 --> 00:09:07,170
to have your notes with you.

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00:09:07,170 --> 00:09:09,300
But the midterms are
also a way for you

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to develop some mechanism so
that you don't actually waste

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too much time on things that
you should be able to do

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00:09:14,160 --> 00:09:16,092
without thinking too much.

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You will be allowed
to cheat sheet,

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because, well, you can
always forget something.

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00:09:20,940 --> 00:09:23,790
And it will be
two-sided letters sheet,

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00:09:23,790 --> 00:09:27,790
and you can practice yourself
as writing as small as you want.

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00:09:27,790 --> 00:09:30,510
And you can put whatever you
want on this cheat sheet.

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All right, the final will
be decided by the register.

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00:09:33,037 --> 00:09:34,620
It's going to be
three hours, and it's

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00:09:34,620 --> 00:09:35,832
going to count for 40%.

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You cannot bring books, but
you can bring your notes.

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Yes.

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AUDIENCE: I noticed
that the midterm dates

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aren't dated in the syllabus.

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00:09:41,704 --> 00:09:43,120
So I wanted to
make sure you know.

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00:09:43,120 --> 00:09:44,453
PHILIPPE RIGOLLET: They are not?

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00:09:44,453 --> 00:09:45,470
AUDIENCE: Yeah--

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00:09:45,470 --> 00:09:45,890
PHILIPPE RIGOLLET:
Oh, yeah, there's

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a "1" that's missing on
both of them, isn't there?

230
00:09:59,030 --> 00:10:03,180
Yeah, let's figure that out.

231
00:10:03,180 --> 00:10:05,510
The syllabus is the true one.

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00:10:05,510 --> 00:10:07,422
The slides are so
that we can discuss,

233
00:10:07,422 --> 00:10:08,880
but the ones that's
on the syllabus

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are the ones that count.

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00:10:09,879 --> 00:10:13,050
And I think they're also posted
on the calendar on Stellar

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00:10:13,050 --> 00:10:14,909
as well.

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00:10:14,909 --> 00:10:15,700
Any other question?

238
00:10:20,890 --> 00:10:23,380
OK, so the pre-reqs here--

239
00:10:23,380 --> 00:10:28,100
and who has looked at the
first problem set already?

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00:10:28,100 --> 00:10:30,940
OK, so those hands
that are raised

241
00:10:30,940 --> 00:10:34,160
realize that there is a true
prerequisite of probability

242
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for this class.

243
00:10:36,300 --> 00:10:40,840
It can be at the level
of 18.600 or 604.1.

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00:10:40,840 --> 00:10:42,060
I should say "B" now.

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00:10:42,060 --> 00:10:44,780
It's two classes.

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00:10:44,780 --> 00:10:48,230
I will require you
to know some calculus

247
00:10:48,230 --> 00:10:51,170
and have some notions
of linear algebra,

248
00:10:51,170 --> 00:10:53,510
such as, what is a matrix,
what is a vector, how

249
00:10:53,510 --> 00:10:55,730
do you multiply those
things together,

250
00:10:55,730 --> 00:10:58,400
some notion of what
orthonormal vectors are.

251
00:11:01,280 --> 00:11:03,350
We'll talk about
eigenvectors and eigenvalues,

252
00:11:03,350 --> 00:11:05,010
but I remind you all of that.

253
00:11:05,010 --> 00:11:07,550
So this is not this
strict pre-req.

254
00:11:07,550 --> 00:11:09,910
But if you've taken
it, for example,

255
00:11:09,910 --> 00:11:12,080
it doesn't hurt to
go back to your notes

256
00:11:12,080 --> 00:11:13,940
when we get closer
to this chapter

257
00:11:13,940 --> 00:11:15,950
on principle-component analysis.

258
00:11:15,950 --> 00:11:19,520
The chapters, as they're listed
in the syllabus, are in order,

259
00:11:19,520 --> 00:11:22,430
so you will see when
it actually comes.

260
00:11:22,430 --> 00:11:24,830
There's no required textbook.

261
00:11:24,830 --> 00:11:29,506
And I know you tend
to not like that.

262
00:11:29,506 --> 00:11:31,880
You like to have your textbook
to know where you're going

263
00:11:31,880 --> 00:11:32,810
and what we're doing.

264
00:11:32,810 --> 00:11:34,670
I'm sorry, it's just this class.

265
00:11:34,670 --> 00:11:36,950
Either I would have to go
to a mathematical statistics

266
00:11:36,950 --> 00:11:38,970
textbook, which
is just too much,

267
00:11:38,970 --> 00:11:43,010
or to go to a more
engineering-type statistics

268
00:11:43,010 --> 00:11:45,380
class, which is just too little.

269
00:11:45,380 --> 00:11:49,160
So hopefully, the
problems will be enough

270
00:11:49,160 --> 00:11:50,690
for you to practice
the recitations.

271
00:11:50,690 --> 00:11:52,940
We'll have some problems
to solve as well.

272
00:11:52,940 --> 00:11:55,520
And the material will
be posted on the slides.

273
00:11:55,520 --> 00:11:57,320
So you should have
everything you need.

274
00:11:57,320 --> 00:11:58,799
There's plenty of
resources online

275
00:11:58,799 --> 00:12:00,590
if you want to expand
on a particular topic

276
00:12:00,590 --> 00:12:03,080
or read it as said
by somebody else.

277
00:12:03,080 --> 00:12:07,690
The book that I
recommend in the syllabus

278
00:12:07,690 --> 00:12:11,510
is this book called All of
Statistics by Wasserman.

279
00:12:11,510 --> 00:12:13,580
Mainly because of the
title, I'm guessing

280
00:12:13,580 --> 00:12:16,700
it has all of it in it.

281
00:12:16,700 --> 00:12:18,530
It's pretty broad.

282
00:12:18,530 --> 00:12:20,030
There's actually not that many.

283
00:12:20,030 --> 00:12:22,910
It's more of an
intro-grad level.

284
00:12:22,910 --> 00:12:27,890
But it's not very deep, but
you see a lot of the overview.

285
00:12:27,890 --> 00:12:29,480
Certainly, what
we're going to cover

286
00:12:29,480 --> 00:12:30,979
will be a subset
of what's in there.

287
00:12:34,190 --> 00:12:35,690
The slides will be
posted on Stellar

288
00:12:35,690 --> 00:12:38,420
before lectures before
we start a new chapter

289
00:12:38,420 --> 00:12:41,960
and after we're done with the
chapter, with the annotations,

290
00:12:41,960 --> 00:12:46,784
and also, with the typos
corrected, like for the exam.

291
00:12:46,784 --> 00:12:48,200
There will be some
video lectures.

292
00:12:48,200 --> 00:12:52,550
Again, the first one will be
posted on OCW from last year.

293
00:12:52,550 --> 00:12:54,950
But all of them will be
available on Stellar--

294
00:12:54,950 --> 00:12:57,770
of course, module
technical problems.

295
00:12:57,770 --> 00:13:00,317
But this is an automated system.

296
00:13:00,317 --> 00:13:02,150
And hopefully, it will
work out well for us.

297
00:13:02,150 --> 00:13:04,490
So if you somehow have
to miss a lecture,

298
00:13:04,490 --> 00:13:08,270
you can always catch
it up by watching it.

299
00:13:08,270 --> 00:13:10,760
You can also play
at that speed 0.75

300
00:13:10,760 --> 00:13:12,530
in case I end up
speaking too fast,

301
00:13:12,530 --> 00:13:15,470
but I think I've
managed myself so far--

302
00:13:15,470 --> 00:13:19,690
so just last warning.

303
00:13:19,690 --> 00:13:22,910
All right, why should
you study statistics?

304
00:13:22,910 --> 00:13:27,362
Well, if you read the news, you
will see a lot of statistics.

305
00:13:27,362 --> 00:13:28,570
I mentioned machine learning.

306
00:13:28,570 --> 00:13:32,269
It's built on a
lot of statistics.

307
00:13:32,269 --> 00:13:34,060
If I were to teach this
class 10 years ago,

308
00:13:34,060 --> 00:13:37,480
I would have to explain to you
that data collection and making

309
00:13:37,480 --> 00:13:40,510
decisions based on data was
something that made sense.

310
00:13:40,510 --> 00:13:43,180
But now, it's
almost in our life.

311
00:13:43,180 --> 00:13:47,470
We're used to this idea that
data helps in making decisions.

312
00:13:47,470 --> 00:13:51,680
And people use data
to conduct studies.

313
00:13:51,680 --> 00:13:55,344
So here, I found a bunch
of press titles that--

314
00:13:55,344 --> 00:13:57,760
I think the key word I was
looking for was "study finds"--

315
00:13:57,760 --> 00:13:58,635
if I want to do this.

316
00:13:58,635 --> 00:14:01,640
So I actually did not bother
doing it again this year.

317
00:14:01,640 --> 00:14:04,540
This is all 2016, 2016, 2016.

318
00:14:04,540 --> 00:14:07,030
But the key word that I look
for is usually "study find"--

319
00:14:07,030 --> 00:14:08,350
so a new study find--

320
00:14:08,350 --> 00:14:10,710
traffic is bad for your health.

321
00:14:10,710 --> 00:14:13,740
So we had to wait for 2016
for data to tell us that.

322
00:14:13,740 --> 00:14:18,490
And there's a bunch of other
slightly more interesting ones.

323
00:14:18,490 --> 00:14:20,530
For example, one that you
might find interesting

324
00:14:20,530 --> 00:14:24,280
is that this study finds that
students benefit from waiting

325
00:14:24,280 --> 00:14:26,000
to declare a major.

326
00:14:26,000 --> 00:14:28,300
Now, there's a bunch
of press titles.

327
00:14:28,300 --> 00:14:33,385
There one in the MIT News
that finds brain connections,

328
00:14:33,385 --> 00:14:34,870
key to reading.

329
00:14:34,870 --> 00:14:37,750
And so here, we have an
idea of what happened there.

330
00:14:37,750 --> 00:14:39,490
Some data was collected.

331
00:14:39,490 --> 00:14:42,580
Some scientific
hypothesis was formulated.

332
00:14:42,580 --> 00:14:47,920
And then the data was here
to try to prove or disprove

333
00:14:47,920 --> 00:14:49,360
this scientific hypothesis.

334
00:14:49,360 --> 00:14:51,600
That's the usual
scientific process.

335
00:14:51,600 --> 00:14:55,300
And we need to understand how
the scientific process goes,

336
00:14:55,300 --> 00:14:58,030
because some of those things
might be actually questionable.

337
00:14:58,030 --> 00:15:02,140
Who is 100% sure that
study finds that students--

338
00:15:02,140 --> 00:15:04,600
do you think that you
benefit from waiting

339
00:15:04,600 --> 00:15:05,710
to declare a major?

340
00:15:05,710 --> 00:15:09,370
Right I would be
skeptical about this.

341
00:15:09,370 --> 00:15:13,360
I would be like, I don't want
to wait to declare a major.

342
00:15:13,360 --> 00:15:15,130
So what kind of
thing can we bring?

343
00:15:15,130 --> 00:15:17,170
Well maybe this
study studied people

344
00:15:17,170 --> 00:15:18,370
that were different from me.

345
00:15:18,370 --> 00:15:21,126
Or maybe the study
finds that this

346
00:15:21,126 --> 00:15:22,750
is beneficial for a
majority of people.

347
00:15:22,750 --> 00:15:23,890
I'm not a majority.

348
00:15:23,890 --> 00:15:24,724
I'm just one person.

349
00:15:24,724 --> 00:15:26,098
There's a bunch
of things that we

350
00:15:26,098 --> 00:15:28,540
need to understand what
those things actually mean.

351
00:15:28,540 --> 00:15:30,248
And we'll see that
those are actually not

352
00:15:30,248 --> 00:15:31,511
statements about individuals.

353
00:15:31,511 --> 00:15:33,760
They're not even statements
about the cohort of people

354
00:15:33,760 --> 00:15:35,050
they've actually looked at.

355
00:15:35,050 --> 00:15:37,270
They're statements
about a parameter

356
00:15:37,270 --> 00:15:40,860
of a distribution
that was used to model

357
00:15:40,860 --> 00:15:43,720
the benefit of waiting.

358
00:15:43,720 --> 00:15:45,040
So there's a lot of questions.

359
00:15:45,040 --> 00:15:46,840
And there are a lot of
layers that come into this.

360
00:15:46,840 --> 00:15:49,131
And we're going to want to
understand what was going on

361
00:15:49,131 --> 00:15:52,992
in there and try to peel it off
and understand what assumptions

362
00:15:52,992 --> 00:15:53,950
have been put in there.

363
00:15:53,950 --> 00:15:59,650
Even though it looks like
a totally legit study, out

364
00:15:59,650 --> 00:16:01,759
of those studies,
statistically, I

365
00:16:01,759 --> 00:16:04,050
think there's going to be
one that's going to be wrong.

366
00:16:04,050 --> 00:16:05,140
Well, maybe not one.

367
00:16:05,140 --> 00:16:07,930
But if I put a
long list of those,

368
00:16:07,930 --> 00:16:10,150
there would be a few that
would actually be wrong.

369
00:16:10,150 --> 00:16:12,840
If I put 20, there would
definitely be one that's wrong.

370
00:16:12,840 --> 00:16:13,840
So you have to see that.

371
00:16:13,840 --> 00:16:16,600
Every time you see 20 studies,
one is probably wrong.

372
00:16:16,600 --> 00:16:19,000
When there are studies
about drug effects,

373
00:16:19,000 --> 00:16:21,260
out of a list of 100,
one would be wrong.

374
00:16:21,260 --> 00:16:23,590
So we'll see what that means
and what I mean by that.

375
00:16:26,840 --> 00:16:30,980
Of course, not only studies
that make discoveries

376
00:16:30,980 --> 00:16:32,930
are actually making
the press titles.

377
00:16:32,930 --> 00:16:36,890
There's also the press
that talks about things

378
00:16:36,890 --> 00:16:38,480
that make no sense.

379
00:16:38,480 --> 00:16:41,600
I love this first experiment--
the salmon experiment.

380
00:16:41,600 --> 00:16:44,240
Actually, it was
a grad student who

381
00:16:44,240 --> 00:16:47,720
came to a neuroscience
poster session,

382
00:16:47,720 --> 00:16:50,780
pulled out this
poster, and explained

383
00:16:50,780 --> 00:16:53,210
the scientific experiment
that he was conducting,

384
00:16:53,210 --> 00:16:59,510
which consisted in taking a
previously frozen and thawed

385
00:16:59,510 --> 00:17:02,060
salmon, putting it
in an MRI, showing it

386
00:17:02,060 --> 00:17:07,760
pictures of violent images, and
recording its brain activity.

387
00:17:07,760 --> 00:17:11,780
And he was able to discover a
few voxels that were activated

388
00:17:11,780 --> 00:17:14,180
by those violent images.

389
00:17:14,180 --> 00:17:16,339
And can somebody tell
me what happened here?

390
00:17:19,200 --> 00:17:23,530
Was the salmon responding
to the violent activity?

391
00:17:23,530 --> 00:17:26,020
Basically, this is just
a statistical fluke.

392
00:17:26,020 --> 00:17:27,730
That's just randomness at play.

393
00:17:27,730 --> 00:17:29,936
There's so many voxels
that are recorded,

394
00:17:29,936 --> 00:17:31,310
and there's so
many fluctuations.

395
00:17:31,310 --> 00:17:32,630
There's always a
little bit of noise

396
00:17:32,630 --> 00:17:34,588
when you're in those
things, that some of them,

397
00:17:34,588 --> 00:17:36,472
just by chance, got lit up.

398
00:17:36,472 --> 00:17:38,680
And so we need to understand
how to correct for that.

399
00:17:38,680 --> 00:17:40,210
In this particular
instance, we need

400
00:17:40,210 --> 00:17:43,600
to have tools that tell us that,
well, finding three voxels that

401
00:17:43,600 --> 00:17:47,020
are activated for
that many voxels

402
00:17:47,020 --> 00:17:50,200
that you can find in
the salmon's brain

403
00:17:50,200 --> 00:17:53,030
is just too small of a number.

404
00:17:53,030 --> 00:17:56,050
Maybe we need to find a clump
of 20 of them, for example.

405
00:17:56,050 --> 00:17:57,670
All right, so
we're going to have

406
00:17:57,670 --> 00:18:02,710
mathematical tools that help us
find those particular numbers.

407
00:18:02,710 --> 00:18:07,780
I don't know if you ever
saw this one by John Oliver

408
00:18:07,780 --> 00:18:11,230
about phacking.

409
00:18:11,230 --> 00:18:14,110
Or actually, it said p-hacking.

410
00:18:14,110 --> 00:18:17,710
Basically, what John
Oliver is saying

411
00:18:17,710 --> 00:18:20,410
is actually a full-length-- like
there's long segments on this.

412
00:18:20,410 --> 00:18:24,220
And he was explaining how
there's a sociology question

413
00:18:24,220 --> 00:18:28,930
here about how there's a
huge incentive for scientists

414
00:18:28,930 --> 00:18:30,005
to publish results.

415
00:18:30,005 --> 00:18:31,630
You're not going to
say, you know what?

416
00:18:31,630 --> 00:18:33,400
This year, I found nothing.

417
00:18:33,400 --> 00:18:35,277
And so people are
trying to find things.

418
00:18:35,277 --> 00:18:36,860
And just by searching,
it's as if they

419
00:18:36,860 --> 00:18:39,070
were searching for all
the voxels in a brain

420
00:18:39,070 --> 00:18:41,920
until they find one that
was just lit up by chance.

421
00:18:41,920 --> 00:18:43,690
And so they just run
all these studies.

422
00:18:43,690 --> 00:18:46,960
And at some point, one will
be right just out of chance.

423
00:18:46,960 --> 00:18:49,630
And so we have to be very
careful about doing this.

424
00:18:49,630 --> 00:18:52,435
There's much more complicated
problems associated

425
00:18:52,435 --> 00:18:53,920
to what's called
p-hacking, which

426
00:18:53,920 --> 00:18:57,970
consists of violating the basic
assumptions, in particular,

427
00:18:57,970 --> 00:19:00,130
looking at the data,
and then formulating

428
00:19:00,130 --> 00:19:02,260
your scientific
assumption based on data,

429
00:19:02,260 --> 00:19:03,400
and then going back to it.

430
00:19:03,400 --> 00:19:04,358
Your idea doesn't work.

431
00:19:04,358 --> 00:19:05,810
Let's just formulate
another one.

432
00:19:05,810 --> 00:19:07,670
And if you are doing
this, all bets are off.

433
00:19:10,127 --> 00:19:11,710
The theory that we're
going to develop

434
00:19:11,710 --> 00:19:14,170
is actually for a very
clean use of data, which

435
00:19:14,170 --> 00:19:16,450
might be a little unpleasant.

436
00:19:16,450 --> 00:19:19,930
If you've had an army of
graduate students collecting

437
00:19:19,930 --> 00:19:21,730
genomic data for a
year, for example,

438
00:19:21,730 --> 00:19:23,920
maybe you don't
want to say, well,

439
00:19:23,920 --> 00:19:25,810
I had one hypothesis
that didn't work.

440
00:19:25,810 --> 00:19:27,940
Let's throw all the
data into the trash.

441
00:19:27,940 --> 00:19:30,650
And so we need to find
ways to be able to do this.

442
00:19:30,650 --> 00:19:34,750
And there's actually a
course been taught at BU.

443
00:19:34,750 --> 00:19:37,180
It's still in its early
stages, but something

444
00:19:37,180 --> 00:19:39,205
called "adaptive data
analysis" that will allow

445
00:19:39,205 --> 00:19:42,615
you to do these kind of things.

446
00:19:42,615 --> 00:19:43,569
Questions?

447
00:19:46,910 --> 00:19:49,870
OK, so of course,
statistics is not

448
00:19:49,870 --> 00:19:52,480
just for you to be
able to read the press.

449
00:19:52,480 --> 00:19:56,350
Statistics will probably
be used in whatever career

450
00:19:56,350 --> 00:19:58,570
path you choose for yourself.

451
00:19:58,570 --> 00:20:03,070
It started in the 10th century
in Netherlands for hydrology.

452
00:20:03,070 --> 00:20:06,410
Netherlands is basically
under water, under sea level.

453
00:20:06,410 --> 00:20:08,237
And so they wanted
to build some dikes.

454
00:20:08,237 --> 00:20:09,820
But once you're going
to build a dike,

455
00:20:09,820 --> 00:20:11,890
you want to make sure
that it's going to sustain

456
00:20:11,890 --> 00:20:13,875
some tides and some floods.

457
00:20:13,875 --> 00:20:15,250
And so in particular,
they wanted

458
00:20:15,250 --> 00:20:19,799
to build dikes that were high
enough, but not too high.

459
00:20:19,799 --> 00:20:21,340
You could always
say, well, I'm going

460
00:20:21,340 --> 00:20:25,750
to build a 500-meter dike,
and then I'm going to be safe.

461
00:20:25,750 --> 00:20:27,580
You want something
that's based on data.

462
00:20:27,580 --> 00:20:28,510
You want to make sure.

463
00:20:28,510 --> 00:20:30,400
And so in particular,
what did they do?

464
00:20:30,400 --> 00:20:33,700
Well, they collected
data for previous floods.

465
00:20:33,700 --> 00:20:36,220
And then they just
found a dike that

466
00:20:36,220 --> 00:20:37,720
was going to cover
all these things.

467
00:20:37,720 --> 00:20:40,060
Now, if you look at the
data they probably had,

468
00:20:40,060 --> 00:20:41,590
maybe it was scarce.

469
00:20:41,590 --> 00:20:43,484
Maybe they had 10 data points.

470
00:20:43,484 --> 00:20:44,900
And so for those
data points, then

471
00:20:44,900 --> 00:20:47,622
maybe they wanted to
sort of interpolate

472
00:20:47,622 --> 00:20:50,080
between those points, maybe
extrapolate for the larger one.

473
00:20:50,080 --> 00:20:51,810
Based on what they've
seen, maybe they

474
00:20:51,810 --> 00:20:53,740
have chances of
seeing something which

475
00:20:53,740 --> 00:20:56,530
is even larger than everything
they've seen before.

476
00:20:56,530 --> 00:21:00,710
And that's exactly the goal
of statistical modeling--

477
00:21:00,710 --> 00:21:04,030
being able to extrapolate
beyond the data that you have,

478
00:21:04,030 --> 00:21:08,050
guessing what you have
not seen yet might happen.

479
00:21:08,050 --> 00:21:10,390
When you buy insurance
for your car,

480
00:21:10,390 --> 00:21:14,050
or your apartment,
or your phone,

481
00:21:14,050 --> 00:21:16,240
there is a premium
that you have to pay.

482
00:21:16,240 --> 00:21:17,800
And this premium
has been determined

483
00:21:17,800 --> 00:21:21,370
based on how much you
are, in expectation, going

484
00:21:21,370 --> 00:21:22,860
to cost the insurance.

485
00:21:22,860 --> 00:21:25,870
It says, OK, this person
has, day a 10% chance

486
00:21:25,870 --> 00:21:28,200
of breaking their iPhone.

487
00:21:28,200 --> 00:21:30,047
An iPhone costs
that much to repair,

488
00:21:30,047 --> 00:21:31,630
so I'm going to
charge them that much.

489
00:21:31,630 --> 00:21:34,690
And then I'm going to add
an extra dollar for my time.

490
00:21:34,690 --> 00:21:36,850
That's basically how those
things are determined.

491
00:21:36,850 --> 00:21:39,220
And so this is using statistics.

492
00:21:39,220 --> 00:21:42,910
This is basically where
statistics is probably

493
00:21:42,910 --> 00:21:43,630
mostly used.

494
00:21:43,630 --> 00:21:45,610
I was personally
trained as an actuary.

495
00:21:45,610 --> 00:21:48,160
And that's me being a
statistician at an insurance

496
00:21:48,160 --> 00:21:50,270
company.

497
00:21:50,270 --> 00:21:55,460
Clinical trials-- this is also
one of the earliest success

498
00:21:55,460 --> 00:21:58,040
stories of statistics.

499
00:21:58,040 --> 00:21:59,570
It's actually now widespread.

500
00:21:59,570 --> 00:22:04,430
Every time a new drug is
approved for market by the FDA,

501
00:22:04,430 --> 00:22:08,570
it requires a very strict
regimen of testing with data,

502
00:22:08,570 --> 00:22:10,407
and control group,
and treatment group,

503
00:22:10,407 --> 00:22:11,990
and how many people
you need in there,

504
00:22:11,990 --> 00:22:17,390
and what kind of significance
you need for those things.

505
00:22:17,390 --> 00:22:19,405
In particular, those
things look like this,

506
00:22:19,405 --> 00:22:21,507
so now it's 5,000 patients.

507
00:22:21,507 --> 00:22:23,090
It depends on what
kind of drug it is,

508
00:22:23,090 --> 00:22:26,630
but for, say, 100
patients, 56 were cured,

509
00:22:26,630 --> 00:22:29,150
and 44 showed no improvement.

510
00:22:29,150 --> 00:22:31,310
Does the FDA consider that
this is a good number?

511
00:22:31,310 --> 00:22:37,970
Do they have a table for how
many patients were cured?

512
00:22:37,970 --> 00:22:39,380
Is there a placebo effect?

513
00:22:39,380 --> 00:22:41,060
Do I need a control
group of people that

514
00:22:41,060 --> 00:22:42,680
are actually getting a placebo?

515
00:22:42,680 --> 00:22:44,130
It's not clear,
all these things.

516
00:22:44,130 --> 00:22:46,462
And so there's a lot of
things to put into place.

517
00:22:46,462 --> 00:22:48,170
And there's a lot of
floating parameters.

518
00:22:48,170 --> 00:22:50,100
So hopefully, we're
going to be able to use

519
00:22:50,100 --> 00:22:52,040
statistical modeling
to shrink it down

520
00:22:52,040 --> 00:22:55,190
to a small number of
parameters to be able to ask

521
00:22:55,190 --> 00:22:56,480
very simple questions.

522
00:22:56,480 --> 00:22:59,630
"Is a drug effective" is
not a mathematical equation.

523
00:22:59,630 --> 00:23:02,420
But "Is p larger than 0.5?"

524
00:23:02,420 --> 00:23:04,411
is a mathematical
question And that's

525
00:23:04,411 --> 00:23:05,910
essentially we're
going to be doing.

526
00:23:05,910 --> 00:23:08,710
We're going to take this, is a
drug effective, to reducing to,

527
00:23:08,710 --> 00:23:12,460
is a variable larger than 0.5?

528
00:23:12,460 --> 00:23:15,040
Now, of course genetics
are using that.

529
00:23:15,040 --> 00:23:19,120
That's typically actually
the same size of data

530
00:23:19,120 --> 00:23:21,100
that you would
see for FMRI data.

531
00:23:21,100 --> 00:23:25,480
So this is actually
a study that I found.

532
00:23:25,480 --> 00:23:29,890
You have about 4,000 cases of
Alzheimer's and 8,000 control.

533
00:23:29,890 --> 00:23:32,700
So people without Alzheimer's--
that's what's called a control.

534
00:23:32,700 --> 00:23:34,480
That's something
just to make sure

535
00:23:34,480 --> 00:23:38,770
that you can see the
difference with people

536
00:23:38,770 --> 00:23:42,610
that are not affected by
either a drug or a disease.

537
00:23:42,610 --> 00:23:46,854
Is the gene APOE associated
with Alzheimer's disease?

538
00:23:46,854 --> 00:23:49,270
Everybody can see why this
would be an important question.

539
00:23:49,270 --> 00:23:50,228
We now have it crisper.

540
00:23:50,228 --> 00:23:52,570
It's targeted to
very specific genes.

541
00:23:52,570 --> 00:23:55,554
If we could edit it, or
knock it down, or knock it

542
00:23:55,554 --> 00:23:57,220
up, or boost it, maybe
we could actually

543
00:23:57,220 --> 00:23:58,178
have an impact on that.

544
00:23:58,178 --> 00:24:00,111
So those are very
important questions,

545
00:24:00,111 --> 00:24:02,360
because we have the technology
to target those things.

546
00:24:02,360 --> 00:24:04,630
But we need the answers
about what those things are.

547
00:24:04,630 --> 00:24:07,557
And there's a bunch
of other questions.

548
00:24:07,557 --> 00:24:09,890
The minute you're going to
talk to biologists about say,

549
00:24:09,890 --> 00:24:10,300
I can do that.

550
00:24:10,300 --> 00:24:11,390
They're going to
say, OK, are there

551
00:24:11,390 --> 00:24:12,880
any other genes
within the genes,

552
00:24:12,880 --> 00:24:15,046
or any particular snips
that I can actually look at?

553
00:24:15,046 --> 00:24:17,084
And they're looking at
very different questions.

554
00:24:17,084 --> 00:24:19,000
And when you start asking
all these questions,

555
00:24:19,000 --> 00:24:22,040
you have to be careful, because
you're reusing your data again.

556
00:24:22,040 --> 00:24:26,920
And it might lead you
to wrong conclusions.

557
00:24:26,920 --> 00:24:29,530
And those are all over
the place, those things.

558
00:24:29,530 --> 00:24:32,170
And that's why they go all
the way to John Oliver talking

559
00:24:32,170 --> 00:24:33,567
about them.

560
00:24:33,567 --> 00:24:35,025
Any questions about
those examples?

561
00:24:37,609 --> 00:24:38,900
So this is really a motivation.

562
00:24:38,900 --> 00:24:40,358
Again, we're not
going to just take

563
00:24:40,358 --> 00:24:46,350
this data set of those cases
and look at them in detail.

564
00:24:46,350 --> 00:24:49,370
So what is common to
all these examples?

565
00:24:49,370 --> 00:24:50,960
Like, why do we have
to use statistics

566
00:24:50,960 --> 00:24:53,120
for all those things?

567
00:24:53,120 --> 00:24:55,650
Well, there's the
randomness of the data.

568
00:24:55,650 --> 00:24:59,300
There's some effect that
we just don't understand--

569
00:24:59,300 --> 00:25:02,750
for example, the randomness
associated with the lining up

570
00:25:02,750 --> 00:25:04,010
of some voxels.

571
00:25:04,010 --> 00:25:06,650
Or the fact that as
far as the insurance

572
00:25:06,650 --> 00:25:09,050
is concerned whether you're
going to break your iPhone

573
00:25:09,050 --> 00:25:10,790
or not is essentially
a coin toss.

574
00:25:10,790 --> 00:25:11,810
Fully, it's biased.

575
00:25:11,810 --> 00:25:14,780
But it's a coin toss.

576
00:25:14,780 --> 00:25:16,770
From the perspective
of the statistician,

577
00:25:16,770 --> 00:25:18,590
those things are
actually random events.

578
00:25:18,590 --> 00:25:20,550
And we need to tame
this randomness,

579
00:25:20,550 --> 00:25:21,800
to understand this randomness.

580
00:25:21,800 --> 00:25:23,722
Is this going to be
a lot of randomness?

581
00:25:23,722 --> 00:25:25,430
Or is it going to be
a little randomness?

582
00:25:25,430 --> 00:25:26,846
Is it going to be
something that's

583
00:25:26,846 --> 00:25:29,715
like, out of their people--

584
00:25:32,990 --> 00:25:35,040
let's see, for example,
for the floods.

585
00:25:35,040 --> 00:25:38,000
Were the floods that I
saw consistently almost

586
00:25:38,000 --> 00:25:40,790
the same size?

587
00:25:40,790 --> 00:25:43,310
It was almost a rounding
error, or they're just

588
00:25:43,310 --> 00:25:44,366
really widespread.

589
00:25:44,366 --> 00:25:45,990
All these things, we
need to understand

590
00:25:45,990 --> 00:25:48,320
so we can understand
how to build those dikes

591
00:25:48,320 --> 00:25:54,390
or how to make decisions
based on those data.

592
00:25:54,390 --> 00:25:58,130
And we need to understand
this randomness.

593
00:25:58,130 --> 00:26:01,550
OK, so the associated
questions to randomness

594
00:26:01,550 --> 00:26:03,110
were actually
hidden in the text.

595
00:26:03,110 --> 00:26:05,680
So we talked about
the notion of average.

596
00:26:05,680 --> 00:26:08,672
Right, so as far as the
insurance is concerned,

597
00:26:08,672 --> 00:26:10,880
they want to know in average
with the probability is.

598
00:26:10,880 --> 00:26:13,970
Like, what is your chance of
actually breaking your iPhone?

599
00:26:13,970 --> 00:26:18,330
And that's what came in
this notion of fair premium.

600
00:26:18,330 --> 00:26:21,472
There's this notion
of quantifying chance.

601
00:26:21,472 --> 00:26:23,430
We don't want to talk
maybe only about average,

602
00:26:23,430 --> 00:26:26,280
maybe you want to cover say
99% percent of the floods.

603
00:26:26,280 --> 00:26:31,920
So we need to know what is
the height of a flood that's

604
00:26:31,920 --> 00:26:34,350
higher than 99% of the floods.

605
00:26:34,350 --> 00:26:36,960
But maybe there's 1%
of them, you know.

606
00:26:36,960 --> 00:26:38,790
When doomsday comes,
doomsday comes.

607
00:26:38,790 --> 00:26:40,470
Right, we're not
going to pay for it.

608
00:26:40,470 --> 00:26:43,187
All right, so that's
most of the floods.

609
00:26:43,187 --> 00:26:45,270
And then there's questions
of significance, right?

610
00:26:45,270 --> 00:26:47,880
So you know I give this
example, a second ago

611
00:26:47,880 --> 00:26:50,070
about clinical trials.

612
00:26:50,070 --> 00:26:51,180
I give you some numbers.

613
00:26:51,180 --> 00:26:55,350
Clearly the drug cured more
people than it did not.

614
00:26:55,350 --> 00:26:58,020
But does it mean that
it's significantly good,

615
00:26:58,020 --> 00:26:59,220
or was this just by chance.

616
00:26:59,220 --> 00:27:01,950
Maybe it's just that these
people just recovered.

617
00:27:01,950 --> 00:27:04,710
It's like you know
curing a common cold.

618
00:27:04,710 --> 00:27:06,840
And you feel like,
oh I got cured.

619
00:27:06,840 --> 00:27:09,540
But it's really you waited five
days and then you got cured.

620
00:27:09,540 --> 00:27:11,910
All right, so there's this
notion of significance,

621
00:27:11,910 --> 00:27:12,780
of variability.

622
00:27:12,780 --> 00:27:15,150
All these things
are actually notions

623
00:27:15,150 --> 00:27:18,270
that describe randomness
and quantify randomness

624
00:27:18,270 --> 00:27:19,560
into simple things.

625
00:27:19,560 --> 00:27:21,630
Randomness is a very
complicated beast.

626
00:27:21,630 --> 00:27:24,390
But we can summarize it into
things that we understand.

627
00:27:24,390 --> 00:27:27,630
Just like I am a
complicated object.

628
00:27:27,630 --> 00:27:29,610
I'm made of molecules,
and made of genes,

629
00:27:29,610 --> 00:27:31,290
and made of very
complicated things.

630
00:27:31,290 --> 00:27:34,890
But I can be summarized as
my name, my email address,

631
00:27:34,890 --> 00:27:37,270
my height and my weight,
and maybe for most of you,

632
00:27:37,270 --> 00:27:39,100
this is basically enough.

633
00:27:39,100 --> 00:27:41,220
You will recognize
me without having

634
00:27:41,220 --> 00:27:45,240
to do a biopsy on me
every time you see me.

635
00:27:45,240 --> 00:27:49,020
All right, so, to
understand randomness

636
00:27:49,020 --> 00:27:51,450
you have to go
through probability.

637
00:27:51,450 --> 00:27:53,370
Probability is the
study of randomness.

638
00:27:53,370 --> 00:27:54,130
That's what it is.

639
00:27:54,130 --> 00:27:57,750
That's what the first sentence
that a lecturer in probability

640
00:27:57,750 --> 00:27:58,890
will say.

641
00:27:58,890 --> 00:28:02,040
And so that's why I need the
pre-requisite, because this

642
00:28:02,040 --> 00:28:04,830
is what we're going to use
to describe the randomness.

643
00:28:04,830 --> 00:28:07,830
We'll see in a second how it
interacts with statistics.

644
00:28:07,830 --> 00:28:10,590
So sometimes, and actually
probably most of the time

645
00:28:10,590 --> 00:28:13,460
throughout your
semester on probability,

646
00:28:13,460 --> 00:28:15,810
randomness was very
well understood.

647
00:28:15,810 --> 00:28:18,420
When you saw a
probability problem, here

648
00:28:18,420 --> 00:28:19,860
was the chance of
this happening,

649
00:28:19,860 --> 00:28:21,720
here was the chance
of that happening.

650
00:28:21,720 --> 00:28:23,820
Maybe you had more
complicated questions

651
00:28:23,820 --> 00:28:26,720
that you had some basic
elements to answer.

652
00:28:26,720 --> 00:28:32,127
For example, the probability
that I have HBO is this much.

653
00:28:32,127 --> 00:28:34,710
And the probability that I watch
Game of Thrones is that much.

654
00:28:34,710 --> 00:28:38,160
And given that I play basketball
what is the probability--

655
00:28:38,160 --> 00:28:39,870
you had all these
crazy questions,

656
00:28:39,870 --> 00:28:42,360
but you were able to build them.

657
00:28:42,360 --> 00:28:45,340
But all the basic numbers
were given to you.

658
00:28:45,340 --> 00:28:48,240
Statistics will be about
finding those basic numbers.

659
00:28:48,240 --> 00:28:51,940
All right so some examples
that you've probably seen

660
00:28:51,940 --> 00:28:55,950
were dice, cards,
roulette, flipping coins.

661
00:28:55,950 --> 00:28:57,700
All of these things
are things that you've

662
00:28:57,700 --> 00:28:59,060
seen in a probability class.

663
00:28:59,060 --> 00:29:00,726
And the reason is
because it's very easy

664
00:29:00,726 --> 00:29:02,680
to describe the probability
of each outcome.

665
00:29:02,680 --> 00:29:05,620
For a die we know that
each face is going

666
00:29:05,620 --> 00:29:07,150
to come with probably 1/6.

667
00:29:07,150 --> 00:29:09,910
Now I'm not going to go into
a debate of whether this

668
00:29:09,910 --> 00:29:12,160
is pure randomness or
this is determinism.

669
00:29:12,160 --> 00:29:14,940
I think as a model
for actual randomness

670
00:29:14,940 --> 00:29:18,220
a die is a pretty good
number, flipping a coin

671
00:29:18,220 --> 00:29:20,170
is a pretty good model.

672
00:29:20,170 --> 00:29:22,600
So those are actually
a good thing.

673
00:29:22,600 --> 00:29:24,650
So the questions that you
would see, for example,

674
00:29:24,650 --> 00:29:26,320
in probabilities
are the following.

675
00:29:26,320 --> 00:29:27,640
I roll one die.

676
00:29:27,640 --> 00:29:31,300
Alice gets $1 if the number
of dots is less than three.

677
00:29:31,300 --> 00:29:35,230
Bob gets $2 if the number
of dots is less than two.

678
00:29:35,230 --> 00:29:37,510
Do you want to be Alice or
Bob given that your role is

679
00:29:37,510 --> 00:29:40,965
actually to make money.

680
00:29:40,965 --> 00:29:43,910
Yeah, you want to be Bob, right?

681
00:29:43,910 --> 00:29:45,410
So let's see why.

682
00:29:45,410 --> 00:29:47,300
So if you look at the
expectation of what

683
00:29:47,300 --> 00:29:48,500
Alice makes.

684
00:29:48,500 --> 00:29:51,140
So let's call it a.

685
00:29:51,140 --> 00:29:56,080
This is $1, with
probability 1/2.

686
00:29:56,080 --> 00:29:59,630
So 3/6, that's 1/2.

687
00:29:59,630 --> 00:30:02,820
And the expectation
of what Bob makes,

688
00:30:02,820 --> 00:30:11,370
this is $2 with probably
2/6 and that's 2/3.

689
00:30:11,370 --> 00:30:13,230
Which is definitely
larger than 1/2.

690
00:30:13,230 --> 00:30:17,109
So Bob's expectations
actually a bit higher.

691
00:30:17,109 --> 00:30:18,900
So those are the kind
of questions that you

692
00:30:18,900 --> 00:30:19,941
may ask with probability.

693
00:30:19,941 --> 00:30:21,930
I described to you
exactly, you use the fact

694
00:30:21,930 --> 00:30:25,337
that the die would get
less than three dots,

695
00:30:25,337 --> 00:30:26,420
with probability one half.

696
00:30:26,420 --> 00:30:27,090
We knew that.

697
00:30:27,090 --> 00:30:29,880
And I didn't have to describe
to you what was going on there.

698
00:30:29,880 --> 00:30:32,460
You didn't have to
collect data about a die.

699
00:30:32,460 --> 00:30:34,620
Same thing, you roll two dice.

700
00:30:34,620 --> 00:30:36,180
You choose a number
between 2 and 12

701
00:30:36,180 --> 00:30:42,100
and you win $100 if you choose
the sum of the two dice.

702
00:30:42,100 --> 00:30:45,052
Which number do you pick?

703
00:30:45,052 --> 00:30:46,812
What?

704
00:30:46,812 --> 00:30:47,312
AUDIENCE: 7.

705
00:30:47,312 --> 00:30:48,186
PHILIPPE RIGOLLET: 7.

706
00:30:48,186 --> 00:30:48,800
Why 7?

707
00:30:48,800 --> 00:30:50,430
AUDIENCE: It's the most likely.

708
00:30:50,430 --> 00:30:52,638
PHILIPPE RIGOLLET: That's
the most likely one, right?

709
00:30:52,638 --> 00:30:56,800
So your gain here will be
$100 times the probability

710
00:30:56,800 --> 00:31:00,220
that the sum of the two
dice, let's say x plus y,

711
00:31:00,220 --> 00:31:03,430
is equal to your little
z where a little z is

712
00:31:03,430 --> 00:31:04,478
the number you pick.

713
00:31:07,830 --> 00:31:10,350
So 7 is the most
likely to happen

714
00:31:10,350 --> 00:31:14,580
and that's the one that
maximizes this function of z.

715
00:31:14,580 --> 00:31:17,112
And for this you need to study
a more complicated function.

716
00:31:17,112 --> 00:31:18,820
But it's a function
that enables two die.

717
00:31:18,820 --> 00:31:21,720
But you can compute the
probability that x plus y

718
00:31:21,720 --> 00:31:26,490
is equal to z, for every
z between 2 and 12.

719
00:31:26,490 --> 00:31:29,010
So you know exactly what
the probabilities are

720
00:31:29,010 --> 00:31:30,930
and that's how you
start probability.

721
00:31:35,450 --> 00:31:38,660
So here that's
exactly what I said.

722
00:31:38,660 --> 00:31:43,550
You have a very simple process
that describes basic events.

723
00:31:43,550 --> 00:31:45,074
Probability 1/6
for each of them.

724
00:31:45,074 --> 00:31:46,490
And then you can
build up on that,

725
00:31:46,490 --> 00:31:49,120
and understand probably of
more complicated events.

726
00:31:49,120 --> 00:31:50,750
You can throw some
money in there.

727
00:31:50,750 --> 00:31:52,780
You can be build functions.

728
00:31:52,780 --> 00:31:56,960
You can do very complicated
things building on that.

729
00:31:56,960 --> 00:31:59,780
Now if I was a
statistician, a statistician

730
00:31:59,780 --> 00:32:01,640
would be the guy who
just arrived on earth,

731
00:32:01,640 --> 00:32:03,473
had never seen a die
and needs to understand

732
00:32:03,473 --> 00:32:05,840
that a die come up with
probably 1/6 on each side.

733
00:32:05,840 --> 00:32:08,030
And the way he would do
it is just to roll the die

734
00:32:08,030 --> 00:32:12,440
until he get some counts
and tries to estimate those.

735
00:32:12,440 --> 00:32:14,060
And maybe that guy
would come and say,

736
00:32:14,060 --> 00:32:16,440
well, you know,
actually, the probability

737
00:32:16,440 --> 00:32:23,500
that I get a 1 is 1/6 plus
0.001 and the probability

738
00:32:23,500 --> 00:32:27,650
that I get a 2 is
1/6 minus 0.005.

739
00:32:27,650 --> 00:32:29,910
And there would be some
fluctuations around this.

740
00:32:29,910 --> 00:32:31,868
And it's going to be his
role as a statistician

741
00:32:31,868 --> 00:32:33,500
to say, listen, this
is too complicated

742
00:32:33,500 --> 00:32:34,880
of a model for this thing.

743
00:32:34,880 --> 00:32:36,694
And these should all
be the same numbers.

744
00:32:36,694 --> 00:32:39,110
Just looking at data, they
should be all the same numbers.

745
00:32:39,110 --> 00:32:40,150
And that's part of the modeling.

746
00:32:40,150 --> 00:32:41,930
You make some
simplifying assumptions

747
00:32:41,930 --> 00:32:46,590
that essentially make your
questions more accurate.

748
00:32:46,590 --> 00:32:48,230
Now, of course, if
your model is wrong,

749
00:32:48,230 --> 00:32:50,120
if it's not true that
all the faces arrive

750
00:32:50,120 --> 00:32:54,410
with the same probability, then
you have a model error here.

751
00:32:54,410 --> 00:32:56,146
So we will be
making model errors.

752
00:32:56,146 --> 00:32:57,770
But that's going to
be the price to pay

753
00:32:57,770 --> 00:32:59,645
to be able to extract
anything from our data.

754
00:33:02,860 --> 00:33:07,299
So for more
complicated processes,

755
00:33:07,299 --> 00:33:09,840
so of course nobody's going to
waste their time rolling dice.

756
00:33:09,840 --> 00:33:11,370
I mean, I'm sure
you might have done

757
00:33:11,370 --> 00:33:13,860
this in AP stat or something.

758
00:33:13,860 --> 00:33:18,070
But the need is to estimate
parameters from data.

759
00:33:18,070 --> 00:33:19,830
All right, so for more
complicated things

760
00:33:19,830 --> 00:33:27,350
you might want to estimate
some density parameter

761
00:33:27,350 --> 00:33:29,400
on a particular set of material.

762
00:33:29,400 --> 00:33:31,620
And for this maybe you need
to beam something to it,

763
00:33:31,620 --> 00:33:33,449
and measure how fast
it's coming back.

764
00:33:33,449 --> 00:33:35,490
And you're going to have
some measurement errors.

765
00:33:35,490 --> 00:33:37,281
And maybe you need to
do that several times

766
00:33:37,281 --> 00:33:39,750
and you have a model for
the physical process that's

767
00:33:39,750 --> 00:33:40,960
actually going on.

768
00:33:40,960 --> 00:33:42,750
And physics is usually
a very good way

769
00:33:42,750 --> 00:33:46,560
to get models for
engineering perspective.

770
00:33:46,560 --> 00:33:49,320
But there's models
for sociology where we

771
00:33:49,320 --> 00:33:52,260
have no physical system, right.

772
00:33:52,260 --> 00:33:53,670
God knows how people interact.

773
00:33:53,670 --> 00:33:55,650
And maybe I'm going
to say that the way

774
00:33:55,650 --> 00:33:59,870
I make friends is by first
flipping a coin in my pocket.

775
00:33:59,870 --> 00:34:01,560
And with probability
2/3, I'm going

776
00:34:01,560 --> 00:34:02,730
to make my friend at work.

777
00:34:02,730 --> 00:34:04,410
And with probability
1/3 I'm going

778
00:34:04,410 --> 00:34:05,850
to make my friend at soccer.

779
00:34:05,850 --> 00:34:07,850
And once I make my
friends at soccer--

780
00:34:07,850 --> 00:34:09,750
I decide to make
my friend soccer.

781
00:34:09,750 --> 00:34:11,400
Then I will face
someone who's flipping

782
00:34:11,400 --> 00:34:14,780
the same coin with maybe be
slightly different parameters.

783
00:34:14,780 --> 00:34:16,230
But those things actually exist.

784
00:34:16,230 --> 00:34:18,750
There's models about how
friendships are formed.

785
00:34:18,750 --> 00:34:22,020
And the one I
described is called

786
00:34:22,020 --> 00:34:23,177
the mixed-membership model.

787
00:34:23,177 --> 00:34:25,260
So those are models that
are sort of hypothesized.

788
00:34:25,260 --> 00:34:29,250
And they're more reasonable
than taking into account

789
00:34:29,250 --> 00:34:31,650
all the things that made
you meet that person

790
00:34:31,650 --> 00:34:34,409
at that particular time.

791
00:34:34,409 --> 00:34:38,580
So the goal here--
so based on data now,

792
00:34:38,580 --> 00:34:41,370
once we have the model is going
to be reduced to maybe two,

793
00:34:41,370 --> 00:34:42,989
three, four
parameters, depending

794
00:34:42,989 --> 00:34:44,500
on how complex the model is.

795
00:34:44,500 --> 00:34:48,060
And then your goal will be
to estimate those parameters.

796
00:34:48,060 --> 00:34:51,610
So sometimes the randomness
we have here is real.

797
00:34:51,610 --> 00:34:56,940
So there's some true
randomness in some surveys.

798
00:34:56,940 --> 00:34:58,740
If I pick a random
student, as long

799
00:34:58,740 --> 00:35:00,870
as I believe that my random
number generator that

800
00:35:00,870 --> 00:35:04,200
will pick your random
ID is actually random,

801
00:35:04,200 --> 00:35:06,180
there is something
random about you.

802
00:35:06,180 --> 00:35:07,950
The student that
I pick at random

803
00:35:07,950 --> 00:35:09,390
will be a random student.

804
00:35:09,390 --> 00:35:13,540
The person that I call on
the phone is a random person.

805
00:35:13,540 --> 00:35:16,380
So there's some randomness
that I can build into my system

806
00:35:16,380 --> 00:35:20,310
by drawing something from
a random number generator.

807
00:35:20,310 --> 00:35:22,512
A biased coin is a random thing.

808
00:35:22,512 --> 00:35:24,220
It's not a very
interesting random thing.

809
00:35:24,220 --> 00:35:26,100
But it is a random thing.

810
00:35:26,100 --> 00:35:29,040
Again, if I wash out the
fact that it actually

811
00:35:29,040 --> 00:35:30,510
is a deterministic mechanism.

812
00:35:30,510 --> 00:35:33,230
But at a certain accuracy,
a certain granularity,

813
00:35:33,230 --> 00:35:36,090
this can be thought of as
a truly random experiment.

814
00:35:36,090 --> 00:35:39,180
Measurement error for example,
if you by some measurement

815
00:35:39,180 --> 00:35:39,850
device.

816
00:35:39,850 --> 00:35:42,760
or some optics
device, for example.

817
00:35:42,760 --> 00:35:45,020
You will have like standard
deviation and things that

818
00:35:45,020 --> 00:35:46,080
come on the side of the box.

819
00:35:46,080 --> 00:35:48,621
And it tells you, this will be
making some measurement error.

820
00:35:48,621 --> 00:35:51,480
And it's usually thermal noise
maybe, or things like this.

821
00:35:51,480 --> 00:35:54,570
And those are very
accurately described

822
00:35:54,570 --> 00:35:56,430
by some random phenomenon.

823
00:35:56,430 --> 00:36:01,260
But sometimes, and I'd say most
times, there's no randomness.

824
00:36:01,260 --> 00:36:02,520
There's no randomness.

825
00:36:02,520 --> 00:36:06,810
It's not like you breaking
your iPhone is a random event.

826
00:36:06,810 --> 00:36:09,390
This is just something
that we sweep--

827
00:36:09,390 --> 00:36:11,940
randomness is a big rug
under which we sweep

828
00:36:11,940 --> 00:36:13,650
everything we don't understand.

829
00:36:13,650 --> 00:36:15,900
And we just hope
that in average we've

830
00:36:15,900 --> 00:36:18,911
captured, the average
effect of what's going on.

831
00:36:18,911 --> 00:36:20,910
And the rest of it might
fluctuate to the right,

832
00:36:20,910 --> 00:36:22,470
might fluctuate to the left.

833
00:36:22,470 --> 00:36:26,010
But what remains is
just sort of randomness

834
00:36:26,010 --> 00:36:27,660
that can be averaged out.

835
00:36:27,660 --> 00:36:31,170
So, of course, this is
where the leap of faith is.

836
00:36:31,170 --> 00:36:33,480
We do not know whether we
were correct of doing this.

837
00:36:33,480 --> 00:36:35,957
Maybe we make some
huge systematic biases

838
00:36:35,957 --> 00:36:36,540
by doing this.

839
00:36:36,540 --> 00:36:39,330
Maybe we forget a very
important component.

840
00:36:39,330 --> 00:36:42,450
Right, for example, if I have--

841
00:36:42,450 --> 00:36:45,816
I don't know, let's
think of something--

842
00:36:49,220 --> 00:36:51,080
a drug for breast cancer.

843
00:36:51,080 --> 00:36:52,880
All right, and I
throw out the fact

844
00:36:52,880 --> 00:36:55,640
that my patient is
either a man or woman.

845
00:36:55,640 --> 00:36:57,530
I'm going to have some
serious model biases.

846
00:36:57,530 --> 00:36:58,030
Right.

847
00:36:58,030 --> 00:37:00,645
So if I say I'm going to
collect a random and patient.

848
00:37:00,645 --> 00:37:02,270
And said I'm going
to start doing this.

849
00:37:02,270 --> 00:37:04,790
There's some information
that I really need, clearly,

850
00:37:04,790 --> 00:37:06,880
to build into my model.

851
00:37:06,880 --> 00:37:10,730
And so the model should be
complicated enough, but not too

852
00:37:10,730 --> 00:37:11,420
complicated.

853
00:37:11,420 --> 00:37:13,220
Right so it should take
into account things

854
00:37:13,220 --> 00:37:17,375
there will systematically
be important.

855
00:37:17,375 --> 00:37:19,700
So, in particular, the
simple rule of thumb

856
00:37:19,700 --> 00:37:24,620
is, when you have a
complicated process,

857
00:37:24,620 --> 00:37:26,780
you can think of it as
being a simple process

858
00:37:26,780 --> 00:37:28,205
and some random noise.

859
00:37:28,205 --> 00:37:30,320
Now, again, the random
noise is everything

860
00:37:30,320 --> 00:37:33,320
you don't understand about
the complicated process.

861
00:37:33,320 --> 00:37:37,810
And the simple process is
everything you actually do.

862
00:37:37,810 --> 00:37:40,600
So good modeling,
and this is not

863
00:37:40,600 --> 00:37:43,920
where we'll be
seeing in this class,

864
00:37:43,920 --> 00:37:46,890
consistent choosing
plausible simple models.

865
00:37:46,890 --> 00:37:50,241
And this requires a tremendous
amount of domain knowledge.

866
00:37:50,241 --> 00:37:52,240
And that's why we're not
doing it in this class.

867
00:37:52,240 --> 00:37:54,240
This is not something where I
can make a blanket statement

868
00:37:54,240 --> 00:37:55,620
about making good modeling.

869
00:37:55,620 --> 00:37:58,020
You need to know, if I
were a statistician working

870
00:37:58,020 --> 00:38:00,990
on a study, I would have to
grill the person in front

871
00:38:00,990 --> 00:38:04,840
of me, the expert, for two hours
to know, but how about this?

872
00:38:04,840 --> 00:38:05,970
How about that?

873
00:38:05,970 --> 00:38:06,990
How does this work?

874
00:38:06,990 --> 00:38:08,940
So it requires to
understand a lot of things.

875
00:38:08,940 --> 00:38:14,070
There's this famous statistician
to whom this sentence is

876
00:38:14,070 --> 00:38:16,640
attributed, and it's
probably not his then,

877
00:38:16,640 --> 00:38:21,191
but Tukey said that he
loves being a statistician,

878
00:38:21,191 --> 00:38:23,190
because you get to play
in everybody's backyard.

879
00:38:23,190 --> 00:38:25,440
Right, so you get to
go and see people.

880
00:38:25,440 --> 00:38:28,260
And you get to understand, at
least to a certain extent, what

881
00:38:28,260 --> 00:38:29,430
their problems are.

882
00:38:29,430 --> 00:38:31,110
Enough that you
can actually build

883
00:38:31,110 --> 00:38:33,400
a reasonable model for what
they're actually doing.

884
00:38:33,400 --> 00:38:34,740
So you get to do some sociology.

885
00:38:34,740 --> 00:38:35,865
You get to do some biology.

886
00:38:35,865 --> 00:38:37,167
You get to do some engineering.

887
00:38:37,167 --> 00:38:39,000
And you get to do a lot
of different things.

888
00:38:39,000 --> 00:38:40,625
Right, so he was
actually at some point

889
00:38:40,625 --> 00:38:46,180
predicting the
presidential election.

890
00:38:46,180 --> 00:38:48,720
So, you see, you get to do
a lot of different things.

891
00:38:48,720 --> 00:38:50,850
But it requires a lot
of time to understand

892
00:38:50,850 --> 00:38:52,230
what problem you're working on.

893
00:38:52,230 --> 00:38:54,480
And if you have a particular
application in mind

894
00:38:54,480 --> 00:38:56,644
you're the best person to
actually understand this.

895
00:38:56,644 --> 00:38:58,560
So I'm just going to
give you the basic tools.

896
00:39:07,660 --> 00:39:11,705
So this is the circle of trust.

897
00:39:11,705 --> 00:39:14,010
No, this is really
just a simple graphic

898
00:39:14,010 --> 00:39:15,840
that tells you what's going on.

899
00:39:15,840 --> 00:39:19,170
When you do probability,
you're given the truth.

900
00:39:19,170 --> 00:39:24,330
Somebody tells you what
die God is rolling.

901
00:39:24,330 --> 00:39:27,490
So you know exactly what the
parameters of the problems are.

902
00:39:27,490 --> 00:39:29,490
And what you're trying
to do is to describe what

903
00:39:29,490 --> 00:39:31,410
the outcomes are going to be.

904
00:39:31,410 --> 00:39:34,260
You can say, if you're
rolling a fair die,

905
00:39:34,260 --> 00:39:36,480
you're going to have 1/6
of the time in your data

906
00:39:36,480 --> 00:39:37,699
you're going to have one.

907
00:39:37,699 --> 00:39:39,740
1/6 of the time you're
going to have to have two.

908
00:39:39,740 --> 00:39:42,156
And so you can describe-- if
I told you what the truth is,

909
00:39:42,156 --> 00:39:44,340
you could actually
go into a computer,

910
00:39:44,340 --> 00:39:46,080
either generate some data.

911
00:39:46,080 --> 00:39:51,030
Or you could describe to me
some more macro properties

912
00:39:51,030 --> 00:39:52,350
of what the data would be like.

913
00:39:52,350 --> 00:39:54,600
Oh, I would see a
bunch of numbers

914
00:39:54,600 --> 00:39:57,150
that would be centered
around 35, if I

915
00:39:57,150 --> 00:40:00,182
drew from a Gaussian
distribution centered at 35.

916
00:40:00,182 --> 00:40:01,890
Right, you would know
this kind of thing.

917
00:40:01,890 --> 00:40:07,400
I would know that it's very
unlikely that if my Gaussian

918
00:40:07,400 --> 00:40:08,940
has standard deviation--

919
00:40:08,940 --> 00:40:13,830
is centered on 0, say,
with standard deviation 3.

920
00:40:13,830 --> 00:40:17,380
It's very unlikely that I will
see numbers below minus 10

921
00:40:17,380 --> 00:40:18,750
in above 10, right?

922
00:40:18,750 --> 00:40:21,060
You know this, that you
basically will not see them.

923
00:40:21,060 --> 00:40:25,310
So you know from the truth,
from the distribution

924
00:40:25,310 --> 00:40:27,810
of a random variable that does
not have mu or sigmas, really

925
00:40:27,810 --> 00:40:28,860
numbers there.

926
00:40:28,860 --> 00:40:31,980
You know what data,
you're going to be having.

927
00:40:31,980 --> 00:40:33,895
Statistics is about
going backwards.

928
00:40:33,895 --> 00:40:37,590
It's saying, if I have
some data, what was

929
00:40:37,590 --> 00:40:39,420
the truth that generated it.

930
00:40:39,420 --> 00:40:41,790
And since there are so
many possible truths,

931
00:40:41,790 --> 00:40:44,040
Modeling says you
have to pick one

932
00:40:44,040 --> 00:40:47,190
of the simpler possible truths,
so that you can average out.

933
00:40:47,190 --> 00:40:49,720
Statistics basically
means averaging.

934
00:40:49,720 --> 00:40:51,540
You're averaging when
you do statistics.

935
00:40:51,540 --> 00:40:54,150
And averaging
means that if I say

936
00:40:54,150 --> 00:40:56,790
that I received--
so if I collect

937
00:40:56,790 --> 00:40:58,230
all your GPAs, for example.

938
00:40:58,230 --> 00:41:01,860
And my model is that
the possible GPAs

939
00:41:01,860 --> 00:41:03,450
are any possible numbers.

940
00:41:03,450 --> 00:41:05,297
And anybody can have
any possible GPA.

941
00:41:05,297 --> 00:41:06,880
This is going to be
a serious problem.

942
00:41:06,880 --> 00:41:09,420
But if I can summarize
those GPAs into two numbers,

943
00:41:09,420 --> 00:41:11,100
say, mean and
standard deviation,

944
00:41:11,100 --> 00:41:13,020
than I have a pretty
good description of what

945
00:41:13,020 --> 00:41:15,360
is going on, rather
than having to have

946
00:41:15,360 --> 00:41:16,730
to predict the full list.

947
00:41:16,730 --> 00:41:18,880
Right, if I learn a full
list of GPAs and I say,

948
00:41:18,880 --> 00:41:20,282
well this was the distribution.

949
00:41:20,282 --> 00:41:22,740
Then it's not going to be of
any use for me to predict what

950
00:41:22,740 --> 00:41:25,232
the GPA would be, or some
random student walking in,

951
00:41:25,232 --> 00:41:26,190
or something like this.

952
00:41:30,210 --> 00:41:34,680
So just to finish my rant about
probability versus statistics,

953
00:41:34,680 --> 00:41:37,230
this is a question you would
see in a probability-- this

954
00:41:37,230 --> 00:41:40,080
is a probabilistic question, and
this is a statistical question.

955
00:41:40,080 --> 00:41:42,690
The probabilistic question
is, previous studies

956
00:41:42,690 --> 00:41:45,150
showed that the drug
was 80% effective.

957
00:41:45,150 --> 00:41:46,230
So you know that.

958
00:41:46,230 --> 00:41:48,340
This is the effectiveness
of the drug.

959
00:41:48,340 --> 00:41:49,480
It's given to you.

960
00:41:49,480 --> 00:41:51,480
This is how your problem starts.

961
00:41:51,480 --> 00:41:54,930
Then we can anticipate that,
for a study on 100 patients,

962
00:41:54,930 --> 00:41:57,030
in average, 80 be cured.

963
00:41:57,030 --> 00:42:00,660
And at least 65 will be
cured with 99% chances.

964
00:42:00,660 --> 00:42:03,140
So again these are not--

965
00:42:03,140 --> 00:42:05,994
I'm not predicting on 100
patients exactly the number

966
00:42:05,994 --> 00:42:07,410
of them they're
going to be cured.

967
00:42:07,410 --> 00:42:08,970
And the number of
them that are not.

968
00:42:08,970 --> 00:42:11,160
But I'm actually
sort of predicting

969
00:42:11,160 --> 00:42:13,140
what things are going
to look like on average,

970
00:42:13,140 --> 00:42:17,610
or some macro properties of what
my data sets will look like.

971
00:42:17,610 --> 00:42:19,590
So with 99 percent
chances, that means

972
00:42:19,590 --> 00:42:23,550
that in 99.99% of the
data sets you will

973
00:42:23,550 --> 00:42:25,770
draw from this particular draw.

974
00:42:25,770 --> 00:42:30,870
99.99% of the cohort of 100
patients to whom you administer

975
00:42:30,870 --> 00:42:34,650
this drug, I will be able
to conclude that at least 65

976
00:42:34,650 --> 00:42:41,010
of them will be cured, on 99.99%
percent of those data sets.

977
00:42:41,010 --> 00:42:42,630
So that's a pretty
accurate prediction

978
00:42:42,630 --> 00:42:45,060
of what's going to happen.

979
00:42:45,060 --> 00:42:46,260
Statistics is the opposite.

980
00:42:46,260 --> 00:42:49,620
It says, well, I just know
that 78 out of 100 were cured.

981
00:42:49,620 --> 00:42:50,890
I have only one data set.

982
00:42:50,890 --> 00:42:53,340
I cannot make predictions
for all data sets.

983
00:42:53,340 --> 00:42:57,199
But I can go back
to the probability,

984
00:42:57,199 --> 00:42:59,490
make some inference about
what my probability will look

985
00:42:59,490 --> 00:43:03,360
like, and then say, OK, then
I can make those predictions

986
00:43:03,360 --> 00:43:04,150
later on.

987
00:43:04,150 --> 00:43:08,580
So when I start with
78/100 then maybe

988
00:43:08,580 --> 00:43:11,100
I'm actually, in this
case, I just don't know.

989
00:43:11,100 --> 00:43:16,410
My best guess here is
that I'm confident I

990
00:43:16,410 --> 00:43:19,800
have to add the extra error that
I bet you making by predicting

991
00:43:19,800 --> 00:43:25,650
that here, the drug is not 80%
effective but 78% effective.

992
00:43:25,650 --> 00:43:27,570
And they need some
error bars around this,

993
00:43:27,570 --> 00:43:30,960
that will hopefully contain 80%,
and then based on those error

994
00:43:30,960 --> 00:43:34,460
bars I'm going to make slightly
less precise predictions

995
00:43:34,460 --> 00:43:35,456
for the future.

996
00:43:39,440 --> 00:43:44,510
So, to conclude, so this
was, why statistics?

997
00:43:44,510 --> 00:43:46,530
So what is this course about?

998
00:43:46,530 --> 00:43:48,230
It's about understanding
the mathematics

999
00:43:48,230 --> 00:43:50,230
behind statistical methods.

1000
00:43:50,230 --> 00:43:51,590
It's more of a tool.

1001
00:43:51,590 --> 00:43:54,680
We're not going to have fun and
talk about algebraic geometry

1002
00:43:54,680 --> 00:43:57,440
just for fun in
the middle of it.

1003
00:43:57,440 --> 00:44:01,610
So it justifies quantitative
statements given some modeling

1004
00:44:01,610 --> 00:44:03,500
assumptions, that we
will, in this class,

1005
00:44:03,500 --> 00:44:06,730
mostly admit that the modeling
assumptions are correct.

1006
00:44:06,730 --> 00:44:08,480
| the first part--
in this introduction,

1007
00:44:08,480 --> 00:44:10,442
we will go through
them because it's

1008
00:44:10,442 --> 00:44:12,650
very easy to forget what
the assumptions are actually

1009
00:44:12,650 --> 00:44:13,310
making.

1010
00:44:13,310 --> 00:44:15,650
But this will be a
pretty standard thing.

1011
00:44:15,650 --> 00:44:18,680
The words you will
hear a lot are IID--

1012
00:44:18,680 --> 00:44:20,630
independent and
identically distributed--

1013
00:44:20,630 --> 00:44:23,120
that means that your data
is basically all the sams.

1014
00:44:23,120 --> 00:44:28,070
And one data point is not
impacting another data point.

1015
00:44:28,070 --> 00:44:30,320
Hopefully we can describe
some interesting mathematics

1016
00:44:30,320 --> 00:44:31,400
arising in statistics.

1017
00:44:31,400 --> 00:44:33,500
You know, if you've
taken linear algebra,

1018
00:44:33,500 --> 00:44:36,590
maybe we can explain to you why.

1019
00:44:36,590 --> 00:44:38,630
If you've done some
calculus, maybe we

1020
00:44:38,630 --> 00:44:40,040
can do some
interesting calculus.

1021
00:44:40,040 --> 00:44:42,980
We'll see how in the
spirit of applied math

1022
00:44:42,980 --> 00:44:45,630
those things answer
interesting questions.

1023
00:44:45,630 --> 00:44:49,550
And basically we'll try to
carve out a math toolbox that's

1024
00:44:49,550 --> 00:44:52,300
useful for us statistics.

1025
00:44:52,300 --> 00:44:55,280
And maybe you can extend it
to more sophisticated methods

1026
00:44:55,280 --> 00:44:57,080
that we did not
cover in this class.

1027
00:44:57,080 --> 00:44:59,180
In particular in the
immersion learning class,

1028
00:44:59,180 --> 00:45:02,120
hopefully you'll be able to
have some statistical intuition

1029
00:45:02,120 --> 00:45:04,590
about what is going on.

1030
00:45:04,590 --> 00:45:06,680
So what this course
is not about,

1031
00:45:06,680 --> 00:45:09,950
it's not about spending a lot
of time looking at data sets,

1032
00:45:09,950 --> 00:45:13,310
and trying to understand
some statistical thinking

1033
00:45:13,310 --> 00:45:14,330
kind of questions.

1034
00:45:14,330 --> 00:45:16,550
So this is more of an applied
statistical perspective

1035
00:45:16,550 --> 00:45:19,760
on things, or more modeling.

1036
00:45:19,760 --> 00:45:22,150
So I'm going to typically
give you the model.

1037
00:45:22,150 --> 00:45:23,510
And say this is a model.

1038
00:45:23,510 --> 00:45:26,480
And this is how we're
going to build an estimator

1039
00:45:26,480 --> 00:45:28,440
in the framework of this model.

1040
00:45:28,440 --> 00:45:30,820
So for example, 18.075,
to a certain extent,

1041
00:45:30,820 --> 00:45:32,945
is called "Statistical
Thinking and Data Analysis."

1042
00:45:32,945 --> 00:45:36,500
So I'm hoping there is some
statistical thinking in there.

1043
00:45:36,500 --> 00:45:38,840
We will not talk about
software implementation.

1044
00:45:38,840 --> 00:45:42,270
Unfortunately, there's just
too little time in a semester.

1045
00:45:42,270 --> 00:45:45,840
There's other courses that
are giving you some overview.

1046
00:45:45,840 --> 00:45:49,355
So the main software
these days are R

1047
00:45:49,355 --> 00:45:54,830
is the leading software I'd say
in statistics, both in academia

1048
00:45:54,830 --> 00:45:58,100
and industry, lots of
packages, one every day

1049
00:45:58,100 --> 00:46:00,140
that's probably coming out.

1050
00:46:00,140 --> 00:46:03,410
But there's other things,
right, so now Python is probably

1051
00:46:03,410 --> 00:46:09,110
catching up with all these
scikit-learn packages that

1052
00:46:09,110 --> 00:46:10,130
are coming up.

1053
00:46:10,130 --> 00:46:14,300
Julia has some
statistics in there,

1054
00:46:14,300 --> 00:46:17,867
but it really if you were to
learn a statistical software,

1055
00:46:17,867 --> 00:46:19,325
let's say you love
doing this, this

1056
00:46:19,325 --> 00:46:21,590
would be the one that would
prove most useful for you

1057
00:46:21,590 --> 00:46:22,190
in the future.

1058
00:46:22,190 --> 00:46:26,700
It does not scale super well
to high dimensional data.

1059
00:46:26,700 --> 00:46:28,550
So there is a class
an IDSS that actually

1060
00:46:28,550 --> 00:46:31,080
uses R. It's called
IDS 0.12, I think

1061
00:46:31,080 --> 00:46:36,520
it's called "Statistics,
Computation, and Applications,"

1062
00:46:36,520 --> 00:46:37,850
or something like this.

1063
00:46:37,850 --> 00:46:40,130
I'm also preparing,
with Peter Kempthorne,

1064
00:46:40,130 --> 00:46:42,538
a course called
"Computational Statistics."

1065
00:46:47,380 --> 00:46:50,830
It's going to be offered this
Spring as a special topics.

1066
00:46:50,830 --> 00:46:55,880
And so Peter Kempthorne
will be teaching it.

1067
00:46:55,880 --> 00:46:58,620
And this class
will actually focus

1068
00:46:58,620 --> 00:47:00,807
on using R. And
even beyond that,

1069
00:47:00,807 --> 00:47:02,390
it's not just going
to be about using.

1070
00:47:02,390 --> 00:47:04,285
It's going to be
about understanding--

1071
00:47:04,285 --> 00:47:05,910
just the same way we
we're going to see

1072
00:47:05,910 --> 00:47:07,980
how math helps
you do statistics,

1073
00:47:07,980 --> 00:47:09,870
it's going to help
see how math helps you

1074
00:47:09,870 --> 00:47:12,531
do algorithims for statistics.

1075
00:47:12,531 --> 00:47:15,030
All right, so we'll talk about
maximum likelihood estimator.

1076
00:47:15,030 --> 00:47:16,910
Will need to maximize
some function.

1077
00:47:16,910 --> 00:47:19,120
There's an optimization
toolbox to do that.

1078
00:47:19,120 --> 00:47:20,970
And we'll see how we
can have specialized

1079
00:47:20,970 --> 00:47:22,800
for statistics
for that, and what

1080
00:47:22,800 --> 00:47:25,340
are the principles behind it.

1081
00:47:25,340 --> 00:47:26,990
And you know, of
course, if you've

1082
00:47:26,990 --> 00:47:29,230
taken AP stats you
probably think that stats

1083
00:47:29,230 --> 00:47:31,250
is boring to death
because it was just

1084
00:47:31,250 --> 00:47:34,310
a long laundry-list that
spent a lot of time on t-test.

1085
00:47:34,310 --> 00:47:37,010
I'm pretty sure we're not going
to talk about t-test, well,

1086
00:47:37,010 --> 00:47:38,480
maybe once.

1087
00:47:38,480 --> 00:47:42,200
But this is not a matter of
saying you're going to do this.

1088
00:47:42,200 --> 00:47:43,670
And this is a slight
variant of it.

1089
00:47:43,670 --> 00:47:46,160
We're going to really try to
understand what's going on.

1090
00:47:46,160 --> 00:47:49,820
So, admittedly, you have
not chosen the simplest way

1091
00:47:49,820 --> 00:47:52,580
to get an A in
statistics on campus.

1092
00:47:52,580 --> 00:47:54,470
All right, this is
not the easiest class.

1093
00:47:54,470 --> 00:47:56,000
It might be
challenging at times,

1094
00:47:56,000 --> 00:47:59,674
but I can promise you that
you will maybe suffer.

1095
00:47:59,674 --> 00:48:01,340
But you will learn
something by the time

1096
00:48:01,340 --> 00:48:02,381
you're out of this class.

1097
00:48:02,381 --> 00:48:04,550
This will not be a
waste of your time.

1098
00:48:04,550 --> 00:48:06,560
And you will be
able to understand,

1099
00:48:06,560 --> 00:48:09,230
and not having to remember by
heart how those things actually

1100
00:48:09,230 --> 00:48:10,680
work.

1101
00:48:10,680 --> 00:48:13,750
Are there any questions?

1102
00:48:13,750 --> 00:48:16,910
Anybody want to go to other
stats class on campus?

1103
00:48:16,910 --> 00:48:18,860
Maybe it's not too late.

1104
00:48:18,860 --> 00:48:21,360
OK.

1105
00:48:21,360 --> 00:48:25,280
So let's do some statistics.

1106
00:48:25,280 --> 00:48:29,600
So I see the time
now and it's 11:56,

1107
00:48:29,600 --> 00:48:31,800
so we have another 30 minutes.

1108
00:48:31,800 --> 00:48:35,880
I will typically
give you a three,

1109
00:48:35,880 --> 00:48:37,820
four minute break if
you want to stretch,

1110
00:48:37,820 --> 00:48:39,278
if you want to run
to the bathroom,

1111
00:48:39,278 --> 00:48:45,220
if you want to check
your texts or Instagram.

1112
00:48:45,220 --> 00:48:47,510
There was very little
content in this class,

1113
00:48:47,510 --> 00:48:49,010
hopefully it was
entertaining enough

1114
00:48:49,010 --> 00:48:51,170
that you don't need the break.

1115
00:48:51,170 --> 00:48:55,510
But just in the future, so you
know you will have a break.

1116
00:48:55,510 --> 00:49:01,670
So statistics, this is how it
starts, I'm French, what can

1117
00:49:01,670 --> 00:49:05,750
I say I need to put
some French words.

1118
00:49:05,750 --> 00:49:08,450
So this is not how office
hours are going to go down.

1119
00:49:12,410 --> 00:49:16,370
Anybody know this sculpture
by a Rodin, The Kiss.

1120
00:49:16,370 --> 00:49:18,520
Maybe probably The
Thinker is more famous.

1121
00:49:18,520 --> 00:49:20,240
But this is actually
a pretty famous one.

1122
00:49:20,240 --> 00:49:23,830
But is it really this
one, or is it this one.

1123
00:49:23,830 --> 00:49:26,504
Anybody knows which one it is?

1124
00:49:26,504 --> 00:49:27,910
This one?

1125
00:49:27,910 --> 00:49:28,450
Or this one?

1126
00:49:28,450 --> 00:49:30,628
AUDIENCE: The previous.

1127
00:49:30,628 --> 00:49:32,532
PHILIPPE RIGOLLET: What's that?

1128
00:49:32,532 --> 00:49:33,490
AUDIENCE: This one.

1129
00:49:33,490 --> 00:49:33,990
PHILIPPE RIGOLLET:
It's this one.

1130
00:49:33,990 --> 00:49:35,340
AUDIENCE: Final answer.

1131
00:49:35,340 --> 00:49:39,240
PHILIPPE RIGOLLET: Yeah,
who votes for this one.

1132
00:49:39,240 --> 00:49:40,380
OK.

1133
00:49:40,380 --> 00:49:42,451
Who votes for that one?

1134
00:49:42,451 --> 00:49:42,950
Thank you.

1135
00:49:42,950 --> 00:49:45,770
I love that you do not want to
pronounce yourself with no data

1136
00:49:45,770 --> 00:49:47,160
actually to make any decision.

1137
00:49:47,160 --> 00:49:49,640
This is a total coin toss right.

1138
00:49:49,640 --> 00:49:51,340
Turns out that there
is data, and there

1139
00:49:51,340 --> 00:49:53,930
is in the very serious
journal Nature,

1140
00:49:53,930 --> 00:49:56,210
someone published a
very serious paper which

1141
00:49:56,210 --> 00:49:58,235
actually looks pretty serious.

1142
00:49:58,235 --> 00:50:00,110
If you look at it, it's
like "Human Behavior:

1143
00:50:00,110 --> 00:50:02,510
Adult persistence of
head-turning symmetry,"

1144
00:50:02,510 --> 00:50:04,930
is a lot of fancy
words in there.

1145
00:50:04,930 --> 00:50:07,310
And this, I'm not
kidding you, this study

1146
00:50:07,310 --> 00:50:09,830
is about collecting
data of people kissing,

1147
00:50:09,830 --> 00:50:12,080
and knowing if they bend
their head to the right

1148
00:50:12,080 --> 00:50:14,130
or if they bend they
head to the left.

1149
00:50:14,130 --> 00:50:15,630
And that's all it is.

1150
00:50:15,630 --> 00:50:21,560
And so a neonatal
right-side preference

1151
00:50:21,560 --> 00:50:25,510
makes a surprising romantic
reappearance in later life.

1152
00:50:25,510 --> 00:50:27,830
There's an explanation for it.

1153
00:50:27,830 --> 00:50:32,820
All right, so if we follow this
Nature which one is the one.

1154
00:50:32,820 --> 00:50:33,540
This one?

1155
00:50:33,540 --> 00:50:34,920
Or this one?

1156
00:50:34,920 --> 00:50:35,980
This one, right?

1157
00:50:35,980 --> 00:50:38,900
Head to the right.

1158
00:50:38,900 --> 00:50:41,250
And to be fair, for
this class I was like,

1159
00:50:41,250 --> 00:50:46,130
oh, I'm going to go and show
them what Google Images does.

1160
00:50:46,130 --> 00:50:49,400
When you Google
kissing couple, it's

1161
00:50:49,400 --> 00:50:51,740
inappropriate after
maybe the first picture.

1162
00:50:51,740 --> 00:50:53,540
And so I cannot show you this.

1163
00:50:53,540 --> 00:50:55,370
But you know you can
check for yourself.

1164
00:50:55,370 --> 00:50:57,140
Though I would
argue, so this person

1165
00:50:57,140 --> 00:51:00,890
here actually went
out in airports

1166
00:51:00,890 --> 00:51:06,740
and took pictures of strangers
kissing and collecting data.

1167
00:51:06,740 --> 00:51:10,970
And can somebody guess why
did he just not stay home

1168
00:51:10,970 --> 00:51:12,875
and collect data
from Google Images

1169
00:51:12,875 --> 00:51:17,300
by just googling
kissing couples.

1170
00:51:17,300 --> 00:51:19,850
What's wrong with this data?

1171
00:51:19,850 --> 00:51:22,754
I didn't know actually before I
actually went on Google Images.

1172
00:51:22,754 --> 00:51:23,920
AUDIENCE: It can be altered?

1173
00:51:23,920 --> 00:51:24,540
PHILIPPE RIGOLLET:
What was that?

1174
00:51:24,540 --> 00:51:25,730
AUDIENCE: It can be altered.

1175
00:51:25,730 --> 00:51:26,600
PHILIPPE RIGOLLET:
It can be altered.

1176
00:51:26,600 --> 00:51:28,220
But, you know, who
would want to do this?

1177
00:51:28,220 --> 00:51:29,420
I mean there's no
particular reason why

1178
00:51:29,420 --> 00:51:31,920
you would want to flip an image
before putting it out there.

1179
00:51:31,920 --> 00:51:34,610
I mean, you might, but
you know maybe they

1180
00:51:34,610 --> 00:51:38,858
want to hide the brand of
your Gap shirt or something.

1181
00:51:38,858 --> 00:51:42,260
AUDIENCE: I guess the people
who post pictures of themselves

1182
00:51:42,260 --> 00:51:44,587
kissing on Google Images
are not representative

1183
00:51:44,587 --> 00:51:45,670
of the general population.

1184
00:51:45,670 --> 00:51:47,626
PHILIPPE RIGOLLET:
Yeah, that's very true.

1185
00:51:47,626 --> 00:51:49,250
And actually it's
even worse than that.

1186
00:51:49,250 --> 00:51:51,237
The people who post
pictures of themselves,

1187
00:51:51,237 --> 00:51:52,820
are not posting
pictures of themselves

1188
00:51:52,820 --> 00:51:54,290
or putting pictures
of the people

1189
00:51:54,290 --> 00:51:55,760
that they took a picture of.

1190
00:51:55,760 --> 00:51:59,367
And there usually is a
stock watermark on this.

1191
00:51:59,367 --> 00:52:00,700
And it's basically stock images.

1192
00:52:00,700 --> 00:52:03,440
Those are actors, and so
they've been directed to kiss

1193
00:52:03,440 --> 00:52:06,060
and this is not a
natural thing to do.

1194
00:52:06,060 --> 00:52:08,060
And actually, if you go
to Google Images-- and I

1195
00:52:08,060 --> 00:52:10,602
encourage you to
do this, unless you

1196
00:52:10,602 --> 00:52:12,310
don't want to see
inappropriate pictures,

1197
00:52:12,310 --> 00:52:14,420
and they're mightily
inappropriate.

1198
00:52:14,420 --> 00:52:19,520
And basically you will see
that this study is actually not

1199
00:52:19,520 --> 00:52:20,310
working at all.

1200
00:52:20,310 --> 00:52:21,470
I mean, I looked briefly.

1201
00:52:21,470 --> 00:52:22,920
I didn't actually
collect numbers.

1202
00:52:22,920 --> 00:52:26,460
But I didn't find a particular
tendency to bend right.

1203
00:52:26,460 --> 00:52:28,770
If anything, it was actually
probably the opposite.

1204
00:52:28,770 --> 00:52:31,380
And it's because those people
were directed to do it.

1205
00:52:31,380 --> 00:52:34,975
They just don't actually
think about doing it.

1206
00:52:34,975 --> 00:52:36,350
And also because
I think you need

1207
00:52:36,350 --> 00:52:38,870
to justify writing in
your paper more than,

1208
00:52:38,870 --> 00:52:41,720
I sat in front of my computer.

1209
00:52:41,720 --> 00:52:46,820
So again, this
first sentence here,

1210
00:52:46,820 --> 00:52:49,420
a neonatal right-side
preference--

1211
00:52:49,420 --> 00:52:51,630
"is there a right
side preference?"

1212
00:52:51,630 --> 00:52:53,770
is not a mathematical question.

1213
00:52:53,770 --> 00:52:57,760
But we can start saying, let's
blah, and put some variables,

1214
00:52:57,760 --> 00:52:59,990
and ask questions
about those variables.

1215
00:52:59,990 --> 00:53:02,540
So you know x is actually
not a variable that's

1216
00:53:02,540 --> 00:53:04,870
used very much in
statistics for parameters.

1217
00:53:04,870 --> 00:53:07,129
But p is one, for parameter.

1218
00:53:07,129 --> 00:53:09,420
And so you're going to take
your parameter of interest,

1219
00:53:09,420 --> 00:53:12,100
p, As here is going to be
the proportion of couples.

1220
00:53:12,100 --> 00:53:13,930
And that's among all couples.

1221
00:53:13,930 --> 00:53:17,530
So here, if you talk about
statistical thinking,

1222
00:53:17,530 --> 00:53:20,560
there would be a question about
what population this would

1223
00:53:20,560 --> 00:53:22,080
actually be representative of.

1224
00:53:22,080 --> 00:53:24,190
| usually this is
a call to your--

1225
00:53:26,890 --> 00:53:30,280
sorry, I should not forget this
word it's important for you.

1226
00:53:33,290 --> 00:53:34,720
OK, I forget this word.

1227
00:53:34,720 --> 00:53:38,170
So this is--

1228
00:53:38,170 --> 00:53:43,094
OK,

1229
00:53:43,094 --> 00:53:44,510
So if you look at
this proportion,

1230
00:53:44,510 --> 00:53:46,660
maybe these couples
that are in the study

1231
00:53:46,660 --> 00:53:49,210
might be representative
only of couples in airports.

1232
00:53:49,210 --> 00:53:51,700
Maybe they actually put on a
show for the other passengers.

1233
00:53:51,700 --> 00:53:52,240
Who knows?

1234
00:53:52,240 --> 00:53:54,342
You know, like, oh,
let's just do it as well.

1235
00:53:54,342 --> 00:53:56,050
And just like the
people in Google Images

1236
00:53:56,050 --> 00:53:57,175
they are actually doing it.

1237
00:53:57,175 --> 00:53:58,930
So maybe you want
to just restrict it.

1238
00:53:58,930 --> 00:54:01,120
But of course clearly if
it's appearing in Nature,

1239
00:54:01,120 --> 00:54:04,420
it should not be only
about couples in airports.

1240
00:54:04,420 --> 00:54:07,810
It's supposedly representative
of all couples in the world.

1241
00:54:07,810 --> 00:54:10,210
And so here let's
just keep it vague,

1242
00:54:10,210 --> 00:54:12,610
but you need to keep
in mind what population

1243
00:54:12,610 --> 00:54:14,710
this is actually making
a statement about.

1244
00:54:14,710 --> 00:54:20,120
So you have this full population
of people in the world.

1245
00:54:20,120 --> 00:54:23,130
Right, so those are
all the couples.

1246
00:54:23,130 --> 00:54:27,380
And this person went
ahead and collected data

1247
00:54:27,380 --> 00:54:29,667
about a bunch of them.

1248
00:54:29,667 --> 00:54:31,750
And we know that, in this
thing, there's basically

1249
00:54:31,750 --> 00:54:33,521
a proportion of
them, that's like p,

1250
00:54:33,521 --> 00:54:35,520
and that's the proportion
of them that's bending

1251
00:54:35,520 --> 00:54:36,570
their head to the right.

1252
00:54:36,570 --> 00:54:40,050
And so everybody on this side
is bending their heads right.

1253
00:54:40,050 --> 00:54:41,550
And hopefully we
can actually sample

1254
00:54:41,550 --> 00:54:42,716
this thing you're informing.

1255
00:54:42,716 --> 00:54:44,640
That's basically the
process that's going on.

1256
00:54:44,640 --> 00:54:47,250
So this is the
statistical experiment.

1257
00:54:47,250 --> 00:54:49,589
We're going to observe
n kissing couples.

1258
00:54:49,589 --> 00:54:51,880
So here we're going to put
as many variables as we can.

1259
00:54:51,880 --> 00:54:53,505
So we don't have to
stick with numbers.

1260
00:54:53,505 --> 00:54:55,630
And then we'll just
plug in the numbers.

1261
00:54:55,630 --> 00:54:58,450
n kissing couples, and n
is also, in statistics,

1262
00:54:58,450 --> 00:55:04,320
by the way, n is the size of
your sample 99.9% of the time.

1263
00:55:04,320 --> 00:55:06,720
And collect the value
of each outcome.

1264
00:55:06,720 --> 00:55:07,530
So we want numbers.

1265
00:55:07,530 --> 00:55:08,696
We don't want right or left.

1266
00:55:08,696 --> 00:55:12,300
So we're going to code them
by 0 and 1, pretty naturally.

1267
00:55:12,300 --> 00:55:16,890
And then we're going to
estimate p which is unknown.

1268
00:55:16,890 --> 00:55:18,229
So p is this area.

1269
00:55:18,229 --> 00:55:19,770
And we're going to
estimate it simply

1270
00:55:19,770 --> 00:55:24,570
by the proportion of right
So the proportion of crosses

1271
00:55:24,570 --> 00:55:27,030
that actually fell
in the right side.

1272
00:55:29,660 --> 00:55:33,340
So in this study
what you will find

1273
00:55:33,340 --> 00:55:36,460
is that the numbers
that were collected

1274
00:55:36,460 --> 00:55:43,690
were 124 couples, and that,
out of those 124, 80 of them

1275
00:55:43,690 --> 00:55:46,570
turned their head to the right.

1276
00:55:46,570 --> 00:55:49,240
So, p hat is a proportion.

1277
00:55:49,240 --> 00:55:50,020
How do we do it?

1278
00:55:50,020 --> 00:55:51,880
Well, you don't need
statistics for that.

1279
00:55:51,880 --> 00:55:54,790
You're going to see
80 divided by 124.

1280
00:55:54,790 --> 00:55:57,580
And you will find that
in this particular study

1281
00:55:57,580 --> 00:56:00,139
64.5% of the
couples were bending

1282
00:56:00,139 --> 00:56:01,180
their heads to the right.

1283
00:56:01,180 --> 00:56:03,040
That's a pretty
large number, right?

1284
00:56:03,040 --> 00:56:07,180
The question is if I picked
another 124 couples, maybe

1285
00:56:07,180 --> 00:56:10,240
at different airports, different
times, would I see same number?

1286
00:56:10,240 --> 00:56:11,920
Would this number be
all over the place?

1287
00:56:11,920 --> 00:56:14,650
Would it be sometimes very
close to 120, or sometimes

1288
00:56:14,650 --> 00:56:15,850
for close to 10?

1289
00:56:15,850 --> 00:56:20,020
Or would it be-- is this number
actually fluctuating a lot.

1290
00:56:20,020 --> 00:56:26,870
And so, hopefully not too much,
64.5 percent is definitely

1291
00:56:26,870 --> 00:56:28,670
much larger than 50%.

1292
00:56:28,670 --> 00:56:31,652
And so there seems to
be this preference.

1293
00:56:31,652 --> 00:56:33,110
Now we're going to
have to quantify

1294
00:56:33,110 --> 00:56:34,276
how much of this preference.

1295
00:56:34,276 --> 00:56:38,240
Is this number significantly
larger than 50%?

1296
00:56:38,240 --> 00:56:41,090
So if our data, for example,
was just three couples.

1297
00:56:41,090 --> 00:56:43,370
I'm just going there,
I'm going to Logan.

1298
00:56:43,370 --> 00:56:45,440
I call it, I do
right, left right.

1299
00:56:45,440 --> 00:56:47,570
And then I see--

1300
00:56:47,570 --> 00:56:53,060
see what's the name of
the fish place there?

1301
00:56:53,060 --> 00:56:56,399
I go to I go to Wahlburgers
at Logan and I'm like,

1302
00:56:56,399 --> 00:56:57,440
OK, I'm done for the day.

1303
00:56:57,440 --> 00:56:58,610
I collect this data.

1304
00:56:58,610 --> 00:57:02,120
I go home, and I'm like,
wow, 66.7% to the right.

1305
00:57:02,120 --> 00:57:03,380
That's a pretty big number.

1306
00:57:03,380 --> 00:57:06,650
It's even farther from
50% than this other guy.

1307
00:57:06,650 --> 00:57:08,315
So I'm doing even better.

1308
00:57:08,315 --> 00:57:10,190
But of course you know
that this is not true.

1309
00:57:10,190 --> 00:57:12,237
Three people is definitely
not representative.

1310
00:57:12,237 --> 00:57:13,820
If I stopped at the
first one, I would

1311
00:57:13,820 --> 00:57:18,360
have actually-- at the first
two, I would have even 100%.

1312
00:57:18,360 --> 00:57:21,290
So the question that statistics
is going to help us answer is,

1313
00:57:21,290 --> 00:57:23,514
how large should the sample be?

1314
00:57:23,514 --> 00:57:25,805
For some reason, I don't know
if you guys receive this,

1315
00:57:25,805 --> 00:57:27,554
I'm an affiliate with
the Broad Institute,

1316
00:57:27,554 --> 00:57:30,050
and since then I receive
one email per day

1317
00:57:30,050 --> 00:57:32,240
that says, sample
size determination--

1318
00:57:32,240 --> 00:57:33,950
how large should your sample be?

1319
00:57:33,950 --> 00:57:36,040
Like, I know how large
should with my sample be.

1320
00:57:36,040 --> 00:57:39,530
I've taken 18.650
multiple times.

1321
00:57:39,530 --> 00:57:43,240
And so I know, but the
question is-- is 124

1322
00:57:43,240 --> 00:57:45,270
a large enough number or not?

1323
00:57:45,270 --> 00:57:47,540
Well, the answer is actually,
as usual, it depends.

1324
00:57:47,540 --> 00:57:51,310
It will depend on the
true unknown value of p.

1325
00:57:51,310 --> 00:57:56,590
But from those particular
values that we got, so 120 and--

1326
00:57:56,590 --> 00:57:57,870
how many couples was there?

1327
00:57:57,870 --> 00:57:58,750
80?

1328
00:57:58,750 --> 00:58:02,240
We actually can
make some question.

1329
00:58:02,240 --> 00:58:07,750
So here we said that
80 was larger than 50--

1330
00:58:07,750 --> 00:58:12,310
was allowing us to
conclude at 64.5%.

1331
00:58:12,310 --> 00:58:17,520
So it could be one reason to
say that it was larger than 50%.

1332
00:58:17,520 --> 00:58:23,230
50% of 124 is 62.

1333
00:58:23,230 --> 00:58:24,910
So the question is,
would I be would I

1334
00:58:24,910 --> 00:58:28,750
be willing to make
this conclusion at 63?

1335
00:58:28,750 --> 00:58:30,630
Is that a number that
would convince you?

1336
00:58:30,630 --> 00:58:34,030
Who would be convinced by 63?

1337
00:58:34,030 --> 00:58:35,465
who would be convinced by 72?

1338
00:58:38,410 --> 00:58:40,404
Who would be convinced by 75?

1339
00:58:40,404 --> 00:58:42,820
Hopefully the number of hands
that are raised should grow.

1340
00:58:46,120 --> 00:58:48,820
Who would be convinced by 80?

1341
00:58:48,820 --> 00:58:51,190
All right, so basically
those numbers actually

1342
00:58:51,190 --> 00:58:52,720
don't come from anywhere.

1343
00:58:52,720 --> 00:58:56,740
This 72 would be the number that
you would need for a study--

1344
00:58:56,740 --> 00:58:58,662
most statistical studies
would be the number

1345
00:58:58,662 --> 00:58:59,620
that they would retain.

1346
00:58:59,620 --> 00:59:01,870
That's not for 124.

1347
00:59:01,870 --> 00:59:04,360
You would need to see
72 that turn their head

1348
00:59:04,360 --> 00:59:07,390
right to actually
make this conclusion.

1349
00:59:07,390 --> 00:59:08,977
And then 75--

1350
00:59:08,977 --> 00:59:11,560
So we'll see that there's many
ways to come to this conclusion

1351
00:59:11,560 --> 00:59:12,934
because, as you
can see, this was

1352
00:59:12,934 --> 00:59:15,250
published in Nature with 80.

1353
00:59:15,250 --> 00:59:15,920
So that was OK.

1354
00:59:15,920 --> 00:59:17,503
So 80 is actually a
very large number.

1355
00:59:17,503 --> 00:59:20,116
This is 99 point--

1356
00:59:20,116 --> 00:59:24,250
this 99% -- no, so
this is 95% confidence.

1357
00:59:24,250 --> 00:59:26,290
This is 99% confidence.

1358
00:59:26,290 --> 00:59:29,330
And this is 99.9%
percent confidence.

1359
00:59:29,330 --> 00:59:34,210
So if you said 80 you're a
very conservative person.

1360
00:59:34,210 --> 00:59:36,930
Starting at 72, you can
start making this conclusion.

1361
00:59:39,820 --> 00:59:41,260
To understand
this, we need to do

1362
00:59:41,260 --> 00:59:45,880
our little mathematical
kitchen here,

1363
00:59:45,880 --> 00:59:49,810
and we need to do some modeling.

1364
00:59:49,810 --> 00:59:51,460
So we need to
understand by modeling--

1365
00:59:51,460 --> 00:59:55,000
we need understand what
random process we think

1366
00:59:55,000 --> 00:59:57,190
this data is generating from.

1367
00:59:57,190 --> 00:59:59,180
So it's going to have
some unknown parameters,

1368
00:59:59,180 --> 01:00:00,190
unlike in probability.

1369
01:00:00,190 --> 01:00:02,800
But we need to have just
basically everything written

1370
01:00:02,800 --> 01:00:04,900
except for the values
of the parameters.

1371
01:00:04,900 --> 01:00:08,530
When I said a die is coming
uniformly with probably 1/6

1372
01:00:08,530 --> 01:00:12,220
then I need to have, say
maybe with probability-- maybe

1373
01:00:12,220 --> 01:00:14,260
I should say here
are six numbers,

1374
01:00:14,260 --> 01:00:18,550
and I need to just
fill those numbers.

1375
01:00:18,550 --> 01:00:23,140
So for i equal 1 to
n, I'm going to define

1376
01:00:23,140 --> 01:00:27,140
Ri to be the indicator.

1377
01:00:27,140 --> 01:00:29,950
An indicator is just something
that takes value 1 if something

1378
01:00:29,950 --> 01:00:31,330
is true, and 0 if not.

1379
01:00:31,330 --> 01:00:34,450
So it's an indicator
that i-th couple

1380
01:00:34,450 --> 01:00:36,820
turns the head to the right.

1381
01:00:36,820 --> 01:00:39,580
So, Ri, so it's indexed by i.

1382
01:00:39,580 --> 01:00:42,290
And it's one if the i-th couple
turns their head to the right,

1383
01:00:42,290 --> 01:00:45,820
and 0 if it's--

1384
01:00:45,820 --> 01:00:48,640
well actually, I guess they can
probably kiss straight, right?

1385
01:00:48,640 --> 01:00:51,500
So that would be weird, but
they might be able to do this.

1386
01:00:51,500 --> 01:00:54,280
So let's say not right.

1387
01:00:54,280 --> 01:00:56,710
Then the estimator of
p, we said, was p hat.

1388
01:00:56,710 --> 01:00:58,360
It was just the
ratio of two numbers.

1389
01:00:58,360 --> 01:01:02,410
But really what it is is
I count, I sum those Ri's.

1390
01:01:02,410 --> 01:01:07,360
Since I only add those that
take value 1, what this is is--

1391
01:01:07,360 --> 01:01:10,330
this sum here is actually just
counting the number of 1's.

1392
01:01:10,330 --> 01:01:13,810
Which is another way to say it's
counting the number of couples

1393
01:01:13,810 --> 01:01:15,620
that are kissing to the right.

1394
01:01:15,620 --> 01:01:18,520
And here I don't even
have to tell you anything

1395
01:01:18,520 --> 01:01:20,230
about the numbers or anything.

1396
01:01:20,230 --> 01:01:21,940
I can only keep track of--

1397
01:01:21,940 --> 01:01:24,240
first couple is a 0
second couple is a 1,

1398
01:01:24,240 --> 01:01:25,251
third couple is 0.

1399
01:01:25,251 --> 01:01:27,250
The data set-- you can
actually find it online--

1400
01:01:27,250 --> 01:01:29,646
is actually a sequence
of 0's and 1's.

1401
01:01:29,646 --> 01:01:31,270
Now clearly for the
question that we're

1402
01:01:31,270 --> 01:01:32,950
asking about this
proportion, I don't

1403
01:01:32,950 --> 01:01:34,980
need to keep track of
all this information.

1404
01:01:34,980 --> 01:01:37,150
All I need to keep
track of is the number

1405
01:01:37,150 --> 01:01:39,130
of 0's and the number of 1's.

1406
01:01:39,130 --> 01:01:42,340
Those are completely
interchangeable.

1407
01:01:42,340 --> 01:01:44,440
There's no time effect in this.

1408
01:01:44,440 --> 01:01:48,370
The first couple is no
different than the 15th couple.

1409
01:01:48,370 --> 01:01:50,380
So we call this Rn bar.

1410
01:01:50,380 --> 01:01:52,960
That's going to be a very
standard notation that we use.

1411
01:01:52,960 --> 01:01:55,780
R might be replaced by
other letters like x--

1412
01:01:55,780 --> 01:01:58,150
so xn bar, yn bar.

1413
01:01:58,150 --> 01:02:00,700
And this thing
essentially means that I

1414
01:02:00,700 --> 01:02:04,630
average the R's, or the
Ri's over n of them.

1415
01:02:04,630 --> 01:02:06,100
And the bar means the average.

1416
01:02:06,100 --> 01:02:09,460
So I divide by n the
total number of 1's.

1417
01:02:09,460 --> 01:02:13,630
So here this sum was equal
to 80 in our example and n

1418
01:02:13,630 --> 01:02:16,820
was equal to 124.

1419
01:02:16,820 --> 01:02:18,370
Now this is an estimator.

1420
01:02:18,370 --> 01:02:20,380
So an estimator is
different from an estimate.

1421
01:02:20,380 --> 01:02:21,580
An estimate is a number.

1422
01:02:21,580 --> 01:02:23,680
My estimate was 64.5.

1423
01:02:23,680 --> 01:02:29,502
My estimator is this thing where
I keep all the variables free.

1424
01:02:29,502 --> 01:02:31,210
And in particular, I
keep those variables

1425
01:02:31,210 --> 01:02:34,720
to be random because I'm going
to think of a random couple

1426
01:02:34,720 --> 01:02:37,360
kissing left to right as the
outcome of a random process,

1427
01:02:37,360 --> 01:02:41,320
just like flipping a coin
be getting heads or tails.

1428
01:02:41,320 --> 01:02:43,626
And so this thing here
is a random variable, Ri.

1429
01:02:43,626 --> 01:02:46,250
And this average is, of course,
an average of random variables.

1430
01:02:46,250 --> 01:02:47,500
It's itself a random variable.

1431
01:02:47,500 --> 01:02:49,330
So an estimator is
a random variable.

1432
01:02:49,330 --> 01:02:51,972
An estimate is the realization
of a random variable,

1433
01:02:51,972 --> 01:02:53,680
or, in other words,
is the value that you

1434
01:02:53,680 --> 01:02:56,140
get for this random variable
once you plug in the numbers

1435
01:02:56,140 --> 01:02:58,960
that you've collected.

1436
01:02:58,960 --> 01:03:01,030
So I can talk about the
accuracy of an estimator.

1437
01:03:01,030 --> 01:03:02,470
Accuracy means what?

1438
01:03:02,470 --> 01:03:04,950
Well, what would we
want for an estimator?

1439
01:03:04,950 --> 01:03:06,892
Maybe we won't want it
to fluctuate too much.

1440
01:03:06,892 --> 01:03:07,850
It's a random variable.

1441
01:03:07,850 --> 01:03:11,050
So I'm talking about the
accuracy of a random variable.

1442
01:03:11,050 --> 01:03:13,740
So maybe I don't want
it to be too volatile.

1443
01:03:13,740 --> 01:03:16,440
I could have one
estimator which would be--

1444
01:03:16,440 --> 01:03:20,010
just throw out 182
couples, keep only 2

1445
01:03:20,010 --> 01:03:21,420
and average those two numbers.

1446
01:03:21,420 --> 01:03:23,040
That's definitely
a worse estimator

1447
01:03:23,040 --> 01:03:25,140
than keeping all of the 124.

1448
01:03:25,140 --> 01:03:26,641
So I need to find
a way to say that.

1449
01:03:26,641 --> 01:03:28,140
And what I'm going
to be able to say

1450
01:03:28,140 --> 01:03:30,060
is that the number is
going to be fluctuating.

1451
01:03:30,060 --> 01:03:31,669
If I take another
two couples, I'm

1452
01:03:31,669 --> 01:03:33,210
going to be I'm
probably going to get

1453
01:03:33,210 --> 01:03:34,590
a completely different number.

1454
01:03:34,590 --> 01:03:38,400
But if I take another 124
couples two days later,

1455
01:03:38,400 --> 01:03:40,260
maybe I'm going to have
a very number that's

1456
01:03:40,260 --> 01:03:43,140
very close to 64.5%.

1457
01:03:43,140 --> 01:03:43,969
So that's one way.

1458
01:03:43,969 --> 01:03:46,260
The other thing we would like
about this estimator it's

1459
01:03:46,260 --> 01:03:47,767
actually--

1460
01:03:47,767 --> 01:03:49,350
maybe it's not too
volatile-- but also

1461
01:03:49,350 --> 01:03:54,300
we want it to be close to the
number that we're looking for.

1462
01:03:54,300 --> 01:03:55,410
Here is an estimator.

1463
01:03:55,410 --> 01:03:57,150
It's a beautiful variable.

1464
01:03:57,150 --> 01:04:00,330
72%, that's an estimator.

1465
01:04:00,330 --> 01:04:02,520
Go out there just do
your favorite study

1466
01:04:02,520 --> 01:04:06,310
about drug performance.

1467
01:04:06,310 --> 01:04:10,440
And then they're going to
call you, MIT student taking

1468
01:04:10,440 --> 01:04:12,011
statistics, they
say, so how are you

1469
01:04:12,011 --> 01:04:13,260
going to build your estimator?

1470
01:04:13,260 --> 01:04:15,450
We've collected those 5,000
or something like that.

1471
01:04:15,450 --> 01:04:17,160
I'm just going to spit out 72%.

1472
01:04:17,160 --> 01:04:19,200
Whatever the data says,
that's an estimator.

1473
01:04:19,200 --> 01:04:21,510
It's a stupid estimator
but it is an estimator.

1474
01:04:21,510 --> 01:04:23,360
But this is estimator
is very not volatile.

1475
01:04:23,360 --> 01:04:25,390
Every time you're going
to have a new study,

1476
01:04:25,390 --> 01:04:27,640
even if you change fields,
it's still going to be 72%.

1477
01:04:27,640 --> 01:04:29,190
This is beautiful.

1478
01:04:29,190 --> 01:04:31,590
And the problem is
that's probably not

1479
01:04:31,590 --> 01:04:34,164
very close to the value you're
actually trying to estimate.

1480
01:04:34,164 --> 01:04:35,080
So we need two things.

1481
01:04:35,080 --> 01:04:36,996
We need are estimated
to be a random variable.

1482
01:04:36,996 --> 01:04:39,240
So think in terms of densities.

1483
01:04:39,240 --> 01:04:42,740
We want the density
to be pretty narrow.

1484
01:04:42,740 --> 01:04:46,240
We want this thing
to have very little--

1485
01:04:46,240 --> 01:04:52,570
so this is definitely
better than this.

1486
01:04:52,570 --> 01:04:55,830
But also, we want the number
that we're interested in, p,

1487
01:04:55,830 --> 01:04:57,475
to be very close to this--

1488
01:04:57,475 --> 01:05:00,260
to be close to the values that
this thing is likely to take.

1489
01:05:00,260 --> 01:05:04,020
If p is here, this is
not very good for us.

1490
01:05:04,020 --> 01:05:06,872
So that's basically the things
we're going to be looking at.

1491
01:05:06,872 --> 01:05:08,580
The first one is
referred to as variance.

1492
01:05:08,580 --> 01:05:10,163
The second one is
referred to as bias.

1493
01:05:10,163 --> 01:05:14,380
Those things come all
over in statistics.

1494
01:05:14,380 --> 01:05:16,810
So we need to
understand a model.

1495
01:05:16,810 --> 01:05:20,907
So here's the model that we have
for this particular problem.

1496
01:05:20,907 --> 01:05:22,990
So we need to make assumptions
on the observations

1497
01:05:22,990 --> 01:05:23,650
that we see.

1498
01:05:23,650 --> 01:05:25,620
So we said we're going to assume
that the random variable--

1499
01:05:25,620 --> 01:05:27,160
that's not too much
of a leap of faith.

1500
01:05:27,160 --> 01:05:29,243
We're just sweeping under
the rug everything thing

1501
01:05:29,243 --> 01:05:31,180
we don't understand
about those couples.

1502
01:05:31,180 --> 01:05:33,130
And the assumption
that we make is

1503
01:05:33,130 --> 01:05:36,000
that Ri is a random variable.

1504
01:05:36,000 --> 01:05:38,120
This one you will
forget very soon.

1505
01:05:38,120 --> 01:05:41,950
The second one is that
each of the Ri's is--

1506
01:05:41,950 --> 01:05:45,310
so it's a random variable
that takes value 0 or 1.

1507
01:05:45,310 --> 01:05:47,024
Anybody can suggest
the distribution

1508
01:05:47,024 --> 01:05:48,065
for this random variable?

1509
01:05:48,065 --> 01:05:49,363
AUDIENCE: Bernoulli.

1510
01:05:49,363 --> 01:05:50,363
PHILIPPE RIGOLLET: What?

1511
01:05:50,363 --> 01:05:51,500
AUDIENCE: Bernoulli.

1512
01:05:51,500 --> 01:05:51,980
PHILIPPE RIGOLLET:
Bernoulli, right?

1513
01:05:51,980 --> 01:05:53,210
And it's actually beautiful.

1514
01:05:53,210 --> 01:05:56,770
This is where you have to do
the least statistical modeling.

1515
01:05:56,770 --> 01:05:59,164
A random variable that
takes value 0 or 1

1516
01:05:59,164 --> 01:06:00,080
is always a Bernoulli.

1517
01:06:00,080 --> 01:06:02,581
That's the simplest variable
you can ever think of.

1518
01:06:02,581 --> 01:06:04,580
Any variable that takes
only two possible values

1519
01:06:04,580 --> 01:06:06,155
can be reduced to a Bernoulli.

1520
01:06:06,155 --> 01:06:10,421
OK, so this is a Bernoulli.

1521
01:06:10,421 --> 01:06:12,420
And here we make the
assumption that it actually

1522
01:06:12,420 --> 01:06:16,140
takes parameter p.

1523
01:06:16,140 --> 01:06:17,850
And there's an assumption here.

1524
01:06:17,850 --> 01:06:21,427
Anybody can tell me
what the assumption is?

1525
01:06:21,427 --> 01:06:22,780
AUDIENCE: It's the same.

1526
01:06:22,780 --> 01:06:24,530
PHILIPPE RIGOLLET:
Yeah, it's same, right?

1527
01:06:24,530 --> 01:06:26,194
I could have said
p i, but it's p.

1528
01:06:26,194 --> 01:06:28,110
And that's where I'm
going to be able to start

1529
01:06:28,110 --> 01:06:29,360
getting to do some statistics.

1530
01:06:29,360 --> 01:06:31,930
It's that I'm going to start
to be able to pull information

1531
01:06:31,930 --> 01:06:32,896
across all my guys.

1532
01:06:32,896 --> 01:06:34,270
If I assume that
they're all pi's

1533
01:06:34,270 --> 01:06:36,340
completely uncoupled
with each other.

1534
01:06:36,340 --> 01:06:37,480
Then I'm in trouble.

1535
01:06:37,480 --> 01:06:39,741
There's nothing I
can actually get.

1536
01:06:39,741 --> 01:06:41,740
And then I'm going to
assume that those guys are

1537
01:06:41,740 --> 01:06:42,940
mutually independent.

1538
01:06:42,940 --> 01:06:45,530
And most of the time they
will just say independent.

1539
01:06:45,530 --> 01:06:48,760
Meaning that, it's not like all
these guys called each other

1540
01:06:48,760 --> 01:06:50,110
and it's actually a flash mob.

1541
01:06:50,110 --> 01:06:53,200
And they were like, let's all
turn our left side to the left.

1542
01:06:53,200 --> 01:06:54,790
And then this is
definitely not going

1543
01:06:54,790 --> 01:06:59,910
to give you a valid conclusion.

1544
01:06:59,910 --> 01:07:02,400
So, again. randomness is
a way of modeling lack

1545
01:07:02,400 --> 01:07:03,210
of information.

1546
01:07:03,210 --> 01:07:05,409
Here there is a way
to figure it out.

1547
01:07:05,409 --> 01:07:07,200
Maybe I could have
followed all those guys,

1548
01:07:07,200 --> 01:07:09,040
and knew exactly what
they were-- maybe

1549
01:07:09,040 --> 01:07:11,670
I could have looked at
pictures of them in the womb

1550
01:07:11,670 --> 01:07:14,354
and guess how they were
turning-- by the way that's

1551
01:07:14,354 --> 01:07:16,020
one of the conclusions,
they're guessing

1552
01:07:16,020 --> 01:07:17,436
that we turn our
head to the right

1553
01:07:17,436 --> 01:07:21,180
because our head is turned
to the right in the womb.

1554
01:07:21,180 --> 01:07:24,840
So we don't know what goes
on in the kissers minds.

1555
01:07:24,840 --> 01:07:26,730
And there's, you know,
physics, sociology.

1556
01:07:26,730 --> 01:07:28,521
There's a lot of things
that could help us,

1557
01:07:28,521 --> 01:07:31,220
but it's just too
complicated to keep track of,

1558
01:07:31,220 --> 01:07:34,550
or too expensive
for many instances

1559
01:07:34,550 --> 01:07:37,310
Now again, the nicest
part of this modeling

1560
01:07:37,310 --> 01:07:39,560
was the fact that Ri's
take only two values, which

1561
01:07:39,560 --> 01:07:41,900
mean that this conclusion
that they were Bernoulli

1562
01:07:41,900 --> 01:07:43,220
was totally free for us.

1563
01:07:43,220 --> 01:07:45,530
Once we know it's a random
variable, it's a Bernoulli.

1564
01:07:45,530 --> 01:07:47,720
Now they could have
been, as we said,

1565
01:07:47,720 --> 01:07:51,910
they could have been a
Bernoulli with parameter p i.

1566
01:07:51,910 --> 01:07:55,420
For each i, I could have
put a different parameter,

1567
01:07:55,420 --> 01:07:57,672
but I just don't have
enough information.

1568
01:07:57,672 --> 01:07:58,630
What would I have said?

1569
01:07:58,630 --> 01:08:00,921
I would say, well the first
couple turned to the right.

1570
01:08:00,921 --> 01:08:04,600
p1 has to be one,
that's my best guess.

1571
01:08:04,600 --> 01:08:06,910
The second couple
kiss to the left,

1572
01:08:06,910 --> 01:08:10,690
well, p2 should be 0,
that's my best guess.

1573
01:08:10,690 --> 01:08:14,710
And so basically I
need to have to be

1574
01:08:14,710 --> 01:08:16,240
able to average my information.

1575
01:08:16,240 --> 01:08:19,649
And the way I do it is by
coupling all these guys,

1576
01:08:19,649 --> 01:08:22,490
pi's to be the same p for all i.

1577
01:08:22,490 --> 01:08:23,840
OK, does it make sense?

1578
01:08:23,840 --> 01:08:28,984
Here what I am assuming is that
my population is homogeneous.

1579
01:08:28,984 --> 01:08:29,609
Maybe it's not.

1580
01:08:29,609 --> 01:08:31,484
Maybe I could actually
look at a finer grain,

1581
01:08:31,484 --> 01:08:35,529
but I'm basically making a
statement about a population.

1582
01:08:35,529 --> 01:08:41,670
And so maybe you kiss to the
left, and then you're not--

1583
01:08:41,670 --> 01:08:44,010
I'm not making a statement
about a person individually,

1584
01:08:44,010 --> 01:08:47,702
I'm making a statement about
the overall population.

1585
01:08:47,702 --> 01:08:49,660
Now independence is
probably reasonable, right?

1586
01:08:49,660 --> 01:08:53,080
This person just went
and know can seriously

1587
01:08:53,080 --> 01:08:56,109
hope that these couples did not
communicate with each other.

1588
01:08:56,109 --> 01:08:59,830
Or that you know Tanya did
not text that we should all

1589
01:08:59,830 --> 01:09:01,810
turn our head to the left now.

1590
01:09:01,810 --> 01:09:05,050
And there's no external
stimulus that forces people

1591
01:09:05,050 --> 01:09:08,069
to do something different.

1592
01:09:08,069 --> 01:09:15,810
OK, so-- sorry about that.

1593
01:09:15,810 --> 01:09:19,899
Since we have about
less than 10 minutes.

1594
01:09:19,899 --> 01:09:22,260
Let's do a little bit of
exercises, is that OK with you?

1595
01:09:22,260 --> 01:09:24,950
So I just have some
exercises so we can see what

1596
01:09:24,950 --> 01:09:26,580
an exercise going to look like.

1597
01:09:26,580 --> 01:09:30,090
This is sort of similar to the
exercises you will see with me.

1598
01:09:30,090 --> 01:09:31,674
We should do it together, OK?

1599
01:09:31,674 --> 01:09:32,840
So now we're going to have--

1600
01:09:32,840 --> 01:09:33,479
I have a test.

1601
01:09:36,630 --> 01:09:42,950
So that's an exam
in probability.

1602
01:09:42,950 --> 01:09:44,290
OK.

1603
01:09:44,290 --> 01:09:50,899
And I'm going to have 15
students in this test.

1604
01:09:50,899 --> 01:09:53,550
And hopefully, this
should be 15 grades

1605
01:09:53,550 --> 01:09:57,210
that are representative of the
grades of all a large class.

1606
01:09:57,210 --> 01:10:00,660
Right, so if you go you know
18.600, it's a large class,

1607
01:10:00,660 --> 01:10:02,430
there's definitely
more than 15 students.

1608
01:10:02,430 --> 01:10:04,750
And maybe, just by sampling
15 students at random,

1609
01:10:04,750 --> 01:10:08,010
I want to have an idea of what
my grade distribution will

1610
01:10:08,010 --> 01:10:09,090
look like.

1611
01:10:09,090 --> 01:10:13,230
I'm grading them, I want
to make an educated guess.

1612
01:10:13,230 --> 01:10:15,480
So I'm going to make
some modeling assumptions

1613
01:10:15,480 --> 01:10:16,680
for those guys.

1614
01:10:16,680 --> 01:10:22,830
So here, 15 students and
the grades are x1 to x15.

1615
01:10:22,830 --> 01:10:26,050
Just like we had R1,
R2, all the way to R124.

1616
01:10:26,050 --> 01:10:27,790
Those were my Ri's.

1617
01:10:27,790 --> 01:10:29,610
And so now I have my xi's.

1618
01:10:29,610 --> 01:10:33,270
And I'm going to
assume that xi follows

1619
01:10:33,270 --> 01:10:39,060
a Gaussian or normal
distribution with min mu

1620
01:10:39,060 --> 01:10:40,720
and variance sigma squared.

1621
01:10:40,720 --> 01:10:43,710
Now this is modeling, right?

1622
01:10:43,710 --> 01:10:45,720
Nobody told me there's
no physical process that

1623
01:10:45,720 --> 01:10:46,471
makes this happen.

1624
01:10:46,471 --> 01:10:48,761
We know that there's something
called the central limit

1625
01:10:48,761 --> 01:10:50,720
theorem in the background
that says that things

1626
01:10:50,720 --> 01:10:53,385
tend to be Gaussian, but this is
really a matter of convenience.

1627
01:10:53,385 --> 01:10:55,090
Actually this is, if
you think about it,

1628
01:10:55,090 --> 01:10:57,381
this is terrible because this
puts non-zero probability

1629
01:10:57,381 --> 01:10:58,260
on negative scores.

1630
01:10:58,260 --> 01:11:00,711
I'm definitely not going
to get a negative score.

1631
01:11:00,711 --> 01:11:02,460
But you know it's good
enough because they

1632
01:11:02,460 --> 01:11:05,310
know the probabilities
non-zero but it's probably 10

1633
01:11:05,310 --> 01:11:06,240
to the minus 12.

1634
01:11:06,240 --> 01:11:10,480
So I would be very unlucky
to see a negative score.

1635
01:11:10,480 --> 01:11:24,740
So here's the list of grades, so
I have 65, 41, 70, 90, 58, 82,

1636
01:11:24,740 --> 01:11:28,860
76, 78--

1637
01:11:28,860 --> 01:11:35,660
maybe I should have done
it with 8 --59, 59--

1638
01:11:35,660 --> 01:11:47,360
sitting next to each other
--84, 89, 134, 51, and 72.

1639
01:11:47,360 --> 01:11:51,440
So those are the
scores that I got.

1640
01:11:51,440 --> 01:11:53,930
There were clearly some
bonus points over there.

1641
01:11:53,930 --> 01:12:05,134
And the question is,
find estimator for mu.

1642
01:12:05,134 --> 01:12:06,300
What is my estimator for mu?

1643
01:12:09,120 --> 01:12:11,070
Well, an estimator,
again, is something that

1644
01:12:11,070 --> 01:12:12,510
depends on the random variable.

1645
01:12:12,510 --> 01:12:15,180
All right, so mu is
the expectation, right?

1646
01:12:15,180 --> 01:12:22,510
So a good estimator is
definitely the average score,

1647
01:12:22,510 --> 01:12:24,460
just like we had the
average of the Ri's.

1648
01:12:24,460 --> 01:12:28,210
Now the xi's no longer need
to be 0's and 1's, so it's not

1649
01:12:28,210 --> 01:12:31,060
going to boil down to being
a number of ones divided

1650
01:12:31,060 --> 01:12:32,880
by the total numbers.

1651
01:12:32,880 --> 01:12:41,070
Now if I'm looking
for an estimate,

1652
01:12:41,070 --> 01:12:43,680
well, I need to actually
sum those numbers

1653
01:12:43,680 --> 01:12:45,630
and divide them by 15.

1654
01:12:45,630 --> 01:12:47,472
So my estimate is
going to be 1/15.

1655
01:12:47,472 --> 01:12:49,430
Then I'm going to start
summing those numbers--

1656
01:12:49,430 --> 01:12:51,430
65 plus 72.

1657
01:12:54,320 --> 01:13:06,340
OK, and I can do
it, and it's 67.5.

1658
01:13:06,340 --> 01:13:08,810
This is my estimate.

1659
01:13:08,810 --> 01:13:13,810
Now if I want to compute
a standard deviation--

1660
01:13:13,810 --> 01:13:18,690
so let's say estimate for sigma.

1661
01:13:21,889 --> 01:13:23,180
You've seen that before, right?

1662
01:13:23,180 --> 01:13:24,690
An estimate for sigma is what?

1663
01:13:24,690 --> 01:13:27,150
An estimate for sigma, we'll
see methods to do this,

1664
01:13:27,150 --> 01:13:31,360
but sigma squared
is the variance,

1665
01:13:31,360 --> 01:13:35,941
or is the expectation, of x
minus expectation of x squared.

1666
01:13:38,980 --> 01:13:40,480
And the problem is
that I don't know

1667
01:13:40,480 --> 01:13:42,130
what those expectations are.

1668
01:13:42,130 --> 01:13:47,260
And so I'm going to do what
99.9% percent of statistics is.

1669
01:13:47,260 --> 01:13:49,710
And what is statistics about?

1670
01:13:49,710 --> 01:13:51,350
What's my motto?

1671
01:13:51,350 --> 01:13:54,590
Statistics is about replacing
expectations with averages.

1672
01:13:54,590 --> 01:13:57,020
That's what all of
statistics is about.

1673
01:13:57,020 --> 01:14:00,740
There's 300 pages in a purple
book called All of Statistics

1674
01:14:00,740 --> 01:14:01,940
that tells you this.

1675
01:14:01,940 --> 01:14:03,740
All right, and then
you do something fancy.

1676
01:14:03,740 --> 01:14:05,570
Maybe you minimize
something after you

1677
01:14:05,570 --> 01:14:07,100
replace the expectation.

1678
01:14:07,100 --> 01:14:08,780
Maybe you need to
plug in other stuff.

1679
01:14:08,780 --> 01:14:10,696
But really, every time
you see an expectation,

1680
01:14:10,696 --> 01:14:12,110
you replace it by an average.

1681
01:14:12,110 --> 01:14:13,470
OK let's do this.

1682
01:14:13,470 --> 01:14:16,550
So sigma squared
hat will be what?

1683
01:14:16,550 --> 01:14:20,420
It's going to be 1 over n,
sum from i equals 1 to n

1684
01:14:20,420 --> 01:14:22,790
of xi minus--

1685
01:14:22,790 --> 01:14:25,580
well, here I need to replace
my expectation by an average,

1686
01:14:25,580 --> 01:14:27,230
which is really this average.

1687
01:14:27,230 --> 01:14:31,600
I'm going to call
it mu hat squared.

1688
01:14:31,600 --> 01:14:34,310
There, you have replaced my
expectation with average.

1689
01:14:34,310 --> 01:14:38,150
OK so the golden thing
is, take your expectation

1690
01:14:38,150 --> 01:14:39,460
and replace it with this.

1691
01:14:45,580 --> 01:14:49,240
Frame it, get a tattoo, I don't
care but that's what it is.

1692
01:14:49,240 --> 01:14:53,600
If you remember one thing from
this class, that's what it is.

1693
01:14:53,600 --> 01:14:56,690
Now you can be fancy, if
you look at your calculator,

1694
01:14:56,690 --> 01:14:59,660
it's going to put an n
minus 1 here because it

1695
01:14:59,660 --> 01:15:00,535
wants to be unbiased.

1696
01:15:00,535 --> 01:15:02,410
And those are things we
are going to come to.

1697
01:15:02,410 --> 01:15:04,400
But let's say right
now we stick to this.

1698
01:15:04,400 --> 01:15:06,440
And then when I
plug in my numbers.

1699
01:15:06,440 --> 01:15:14,040
I'm going to get an
estimate for sigma,

1700
01:15:14,040 --> 01:15:17,210
which is the square
root of the estimator

1701
01:15:17,210 --> 01:15:18,810
once I plug in the numbers.

1702
01:15:18,810 --> 01:15:21,622
And you can check that the
number, you get will be 18.

1703
01:15:27,410 --> 01:15:31,060
So those are basic things and
if you've taken any AP stats

1704
01:15:31,060 --> 01:15:32,810
this should be completely
standard to you.

1705
01:15:35,800 --> 01:15:39,350
Now I have another list, and
I don't have time to see it.

1706
01:15:42,029 --> 01:15:43,070
It doesn't really matter.

1707
01:15:49,264 --> 01:15:50,430
OK, we'll do that next time.

1708
01:15:50,430 --> 01:15:51,200
This is fine.

1709
01:15:51,200 --> 01:15:55,241
We'll see another list
of numbers and see--

1710
01:15:55,241 --> 01:15:57,240
we're going to think about
modeling assumptions.

1711
01:15:57,240 --> 01:15:59,610
The goal of this exercise is
not to compute those things,

1712
01:15:59,610 --> 01:16:01,610
it's really to think about
modeling assumptions.

1713
01:16:01,610 --> 01:16:04,002
Is it reasonable to think
that things are IID?

1714
01:16:04,002 --> 01:16:05,460
Is it reasonable
to think that they

1715
01:16:05,460 --> 01:16:07,751
have all the same parameters,
that they're independent,

1716
01:16:07,751 --> 01:16:08,880
et cetera,

1717
01:16:08,880 --> 01:16:16,850
OK so one thing that I wanted
to add is, probably by tonight,

1718
01:16:16,850 --> 01:16:18,810
so I will try to use--

1719
01:16:18,810 --> 01:16:20,625
in the spirit of--

1720
01:16:20,625 --> 01:16:22,250
I don't know what's
starting to happen.

1721
01:16:22,250 --> 01:16:26,190
In the spirit of using
my iPad and fancy things,

1722
01:16:26,190 --> 01:16:29,250
I will try to post some
videos of-- for in particular,

1723
01:16:29,250 --> 01:16:33,840
who has never used a
statistical table to read, say,

1724
01:16:33,840 --> 01:16:37,860
the quantiles of a
Gaussian distribution?

1725
01:16:37,860 --> 01:16:39,480
OK, so there's several of you.

1726
01:16:39,480 --> 01:16:42,600
This is a simple
but boring exercise.

1727
01:16:42,600 --> 01:16:44,850
I will just post a
video on how to do this,

1728
01:16:44,850 --> 01:16:46,785
and you will be able
to find it on Stellar.

1729
01:16:46,785 --> 01:16:48,660
It's going to take five
minutes, and then you

1730
01:16:48,660 --> 01:16:50,770
will know everything there
is to know about those things

1731
01:16:50,770 --> 01:16:53,154
but that's something you need
for the first problem set.

1732
01:16:53,154 --> 01:16:54,570
By the way, so the
problem set has

1733
01:16:54,570 --> 01:16:57,180
30 exercises in probability.

1734
01:16:57,180 --> 01:16:59,510
You need to do 15.

1735
01:16:59,510 --> 01:17:01,470
And you only need to turn in 15.

1736
01:17:01,470 --> 01:17:03,760
You can turn in all
of 30 if you want.

1737
01:17:03,760 --> 01:17:07,230
But you need to know, by the
time we hit those things,

1738
01:17:07,230 --> 01:17:08,760
you need to know--

1739
01:17:08,760 --> 01:17:11,370
well actually, by next week you
need to know what's in there.

1740
01:17:11,370 --> 01:17:13,970
So if you don't have time
to do all the homework,

1741
01:17:13,970 --> 01:17:15,720
and then go back to
your probability class

1742
01:17:15,720 --> 01:17:19,250
to figure out how to do it,
just do 15 easy that you can do.

1743
01:17:19,250 --> 01:17:20,250
And return those things.

1744
01:17:20,250 --> 01:17:21,827
But go back to your
probability class

1745
01:17:21,827 --> 01:17:23,910
and make sure that you
know how to do all of them.

1746
01:17:23,910 --> 01:17:25,650
Those are pretty
basic questions,

1747
01:17:25,650 --> 01:17:28,290
and those are things that I'm
not going to slow down on.

1748
01:17:28,290 --> 01:17:30,900
So you need to remember that
the expectation of the product

1749
01:17:30,900 --> 01:17:32,316
of independent
random variables is

1750
01:17:32,316 --> 01:17:34,560
a product of the expectations.

1751
01:17:34,560 --> 01:17:36,810
Expectation of the sum, is
the sum of the expectation.

1752
01:17:36,810 --> 01:17:38,775
This kind of thing,
which is a little silly,

1753
01:17:38,775 --> 01:17:40,690
but it just requires
you practice.

1754
01:17:40,690 --> 01:17:42,150
So, just have fun.

1755
01:17:42,150 --> 01:17:43,440
Those are simple exercises.

1756
01:17:43,440 --> 01:17:46,620
You will have fun remembering
your probability class.

1757
01:17:46,620 --> 01:17:49,620
All right, so I'll
see you on Tuesday--

1758
01:17:49,620 --> 01:17:51,410
or Monday.