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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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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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I'll tell you a
little bit about what

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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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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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It's one thing to
just bid out a number.

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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,

00:06:42.830 --> 00:06:45.800
so just write them down
on a piece of paper,

00:06:45.800 --> 00:06:48.800
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.

00:08:29.120 --> 00:08:30.941
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.

00:08:50.757 --> 00:08:53.340
PHILIPPE RIGOLLET: We will add
them, multiply the number by 9,

00:08:53.340 --> 00:08:54.298
and that will be grade.

00:08:54.298 --> 00:08:55.410
No.

00:08:55.410 --> 00:08:59.670
I am trying to be nice, there's
just a limit to what I can do.

00:08:59.670 --> 00:09:02.340
All right, so the goal is
for you to learn things

00:09:02.340 --> 00:09:04.020
and to be familiar with them.

00:09:04.020 --> 00:09:05.700
In the final, you
will be allowed

00:09:05.700 --> 00:09:07.170
to have your notes with you.

00:09:07.170 --> 00:09:09.300
But the midterms are
also a way for you

00:09:09.300 --> 00:09:11.760
to develop some mechanism so
that you don't actually waste

00:09:11.760 --> 00:09:14.160
too much time on things that
you should be able to do

00:09:14.160 --> 00:09:16.092
without thinking too much.

00:09:16.092 --> 00:09:17.550
You will be allowed
to cheat sheet,

00:09:17.550 --> 00:09:20.940
because, well, you can
always forget something.

00:09:20.940 --> 00:09:23.790
And it will be
two-sided letters sheet,

00:09:23.790 --> 00:09:27.790
and you can practice yourself
as writing as small as you want.

00:09:27.790 --> 00:09:30.510
And you can put whatever you
want on this cheat sheet.

00:09:30.510 --> 00:09:33.037
All right, the final will
be decided by the register.

00:09:33.037 --> 00:09:34.620
It's going to be
three hours, and it's

00:09:34.620 --> 00:09:35.832
going to count for 40%.

00:09:35.832 --> 00:09:38.040
You cannot bring books, but
you can bring your notes.

00:09:38.040 --> 00:09:38.556
Yes.

00:09:38.556 --> 00:09:40.306
AUDIENCE: I noticed
that the midterm dates

00:09:40.306 --> 00:09:41.704
aren't dated in the syllabus.

00:09:41.704 --> 00:09:43.120
So I wanted to
make sure you know.

00:09:43.120 --> 00:09:44.453
PHILIPPE RIGOLLET: They are not?

00:09:44.453 --> 00:09:45.470
AUDIENCE: Yeah--

00:09:45.470 --> 00:09:45.890
PHILIPPE RIGOLLET:
Oh, yeah, there's

00:09:45.890 --> 00:09:47.973
a "1" that's missing on
both of them, isn't there?

00:09:59.030 --> 00:10:03.180
Yeah, let's figure that out.

00:10:03.180 --> 00:10:05.510
The syllabus is the true one.

00:10:05.510 --> 00:10:07.422
The slides are so
that we can discuss,

00:10:07.422 --> 00:10:08.880
but the ones that's
on the syllabus

00:10:08.880 --> 00:10:09.879
are the ones that count.

00:10:09.879 --> 00:10:13.050
And I think they're also posted
on the calendar on Stellar

00:10:13.050 --> 00:10:14.909
as well.

00:10:14.909 --> 00:10:15.700
Any other question?

00:10:20.890 --> 00:10:23.380
OK, so the pre-reqs here--

00:10:23.380 --> 00:10:28.100
and who has looked at the
first problem set already?

00:10:28.100 --> 00:10:30.940
OK, so those hands
that are raised

00:10:30.940 --> 00:10:34.160
realize that there is a true
prerequisite of probability

00:10:34.160 --> 00:10:36.300
for this class.

00:10:36.300 --> 00:10:40.840
It can be at the level
of 18.600 or 604.1.

00:10:40.840 --> 00:10:42.060
I should say "B" now.

00:10:42.060 --> 00:10:44.780
It's two classes.

00:10:44.780 --> 00:10:48.230
I will require you
to know some calculus

00:10:48.230 --> 00:10:51.170
and have some notions
of linear algebra,

00:10:51.170 --> 00:10:53.510
such as, what is a matrix,
what is a vector, how

00:10:53.510 --> 00:10:55.730
do you multiply those
things together,

00:10:55.730 --> 00:10:58.400
some notion of what
orthonormal vectors are.

00:11:01.280 --> 00:11:03.350
We'll talk about
eigenvectors and eigenvalues,

00:11:03.350 --> 00:11:05.010
but I remind you all of that.

00:11:05.010 --> 00:11:07.550
So this is not this
strict pre-req.

00:11:07.550 --> 00:11:09.910
But if you've taken
it, for example,

00:11:09.910 --> 00:11:12.080
it doesn't hurt to
go back to your notes

00:11:12.080 --> 00:11:13.940
when we get closer
to this chapter

00:11:13.940 --> 00:11:15.950
on principle-component analysis.

00:11:15.950 --> 00:11:19.520
The chapters, as they're listed
in the syllabus, are in order,

00:11:19.520 --> 00:11:22.430
so you will see when
it actually comes.

00:11:22.430 --> 00:11:24.830
There's no required textbook.

00:11:24.830 --> 00:11:29.506
And I know you tend
to not like that.

00:11:29.506 --> 00:11:31.880
You like to have your textbook
to know where you're going

00:11:31.880 --> 00:11:32.810
and what we're doing.

00:11:32.810 --> 00:11:34.670
I'm sorry, it's just this class.

00:11:34.670 --> 00:11:36.950
Either I would have to go
to a mathematical statistics

00:11:36.950 --> 00:11:38.970
textbook, which
is just too much,

00:11:38.970 --> 00:11:43.010
or to go to a more
engineering-type statistics

00:11:43.010 --> 00:11:45.380
class, which is just too little.

00:11:45.380 --> 00:11:49.160
So hopefully, the
problems will be enough

00:11:49.160 --> 00:11:50.690
for you to practice
the recitations.

00:11:50.690 --> 00:11:52.940
We'll have some problems
to solve as well.

00:11:52.940 --> 00:11:55.520
And the material will
be posted on the slides.

00:11:55.520 --> 00:11:57.320
So you should have
everything you need.

00:11:57.320 --> 00:11:58.799
There's plenty of
resources online

00:11:58.799 --> 00:12:00.590
if you want to expand
on a particular topic

00:12:00.590 --> 00:12:03.080
or read it as said
by somebody else.

00:12:03.080 --> 00:12:07.690
The book that I
recommend in the syllabus

00:12:07.690 --> 00:12:11.510
is this book called All of
Statistics by Wasserman.

00:12:11.510 --> 00:12:13.580
Mainly because of the
title, I'm guessing

00:12:13.580 --> 00:12:16.700
it has all of it in it.

00:12:16.700 --> 00:12:18.530
It's pretty broad.

00:12:18.530 --> 00:12:20.030
There's actually not that many.

00:12:20.030 --> 00:12:22.910
It's more of an
intro-grad level.

00:12:22.910 --> 00:12:27.890
But it's not very deep, but
you see a lot of the overview.

00:12:27.890 --> 00:12:29.480
Certainly, what
we're going to cover

00:12:29.480 --> 00:12:30.979
will be a subset
of what's in there.

00:12:34.190 --> 00:12:35.690
The slides will be
posted on Stellar

00:12:35.690 --> 00:12:38.420
before lectures before
we start a new chapter

00:12:38.420 --> 00:12:41.960
and after we're done with the
chapter, with the annotations,

00:12:41.960 --> 00:12:46.784
and also, with the typos
corrected, like for the exam.

00:12:46.784 --> 00:12:48.200
There will be some
video lectures.

00:12:48.200 --> 00:12:52.550
Again, the first one will be
posted on OCW from last year.

00:12:52.550 --> 00:12:54.950
But all of them will be
available on Stellar--

00:12:54.950 --> 00:12:57.770
of course, module
technical problems.

00:12:57.770 --> 00:13:00.317
But this is an automated system.

00:13:00.317 --> 00:13:02.150
And hopefully, it will
work out well for us.

00:13:02.150 --> 00:13:04.490
So if you somehow have
to miss a lecture,

00:13:04.490 --> 00:13:08.270
you can always catch
it up by watching it.

00:13:08.270 --> 00:13:10.760
You can also play
at that speed 0.75

00:13:10.760 --> 00:13:12.530
in case I end up
speaking too fast,

00:13:12.530 --> 00:13:15.470
but I think I've
managed myself so far--

00:13:15.470 --> 00:13:19.690
so just last warning.

00:13:19.690 --> 00:13:22.910
All right, why should
you study statistics?

00:13:22.910 --> 00:13:27.362
Well, if you read the news, you
will see a lot of statistics.

00:13:27.362 --> 00:13:28.570
I mentioned machine learning.

00:13:28.570 --> 00:13:32.269
It's built on a
lot of statistics.

00:13:32.269 --> 00:13:34.060
If I were to teach this
class 10 years ago,

00:13:34.060 --> 00:13:37.480
I would have to explain to you
that data collection and making

00:13:37.480 --> 00:13:40.510
decisions based on data was
something that made sense.

00:13:40.510 --> 00:13:43.180
But now, it's
almost in our life.

00:13:43.180 --> 00:13:47.470
We're used to this idea that
data helps in making decisions.

00:13:47.470 --> 00:13:51.680
And people use data
to conduct studies.

00:13:51.680 --> 00:13:55.344
So here, I found a bunch
of press titles that--

00:13:55.344 --> 00:13:57.760
I think the key word I was
looking for was "study finds"--

00:13:57.760 --> 00:13:58.635
if I want to do this.

00:13:58.635 --> 00:14:01.640
So I actually did not bother
doing it again this year.

00:14:01.640 --> 00:14:04.540
This is all 2016, 2016, 2016.

00:14:04.540 --> 00:14:07.030
But the key word that I look
for is usually "study find"--

00:14:07.030 --> 00:14:08.350
so a new study find--

00:14:08.350 --> 00:14:10.710
traffic is bad for your health.

00:14:10.710 --> 00:14:13.740
So we had to wait for 2016
for data to tell us that.

00:14:13.740 --> 00:14:18.490
And there's a bunch of other
slightly more interesting ones.

00:14:18.490 --> 00:14:20.530
For example, one that you
might find interesting

00:14:20.530 --> 00:14:24.280
is that this study finds that
students benefit from waiting

00:14:24.280 --> 00:14:26.000
to declare a major.

00:14:26.000 --> 00:14:28.300
Now, there's a bunch
of press titles.

00:14:28.300 --> 00:14:33.385
There one in the MIT News
that finds brain connections,

00:14:33.385 --> 00:14:34.870
key to reading.

00:14:34.870 --> 00:14:37.750
And so here, we have an
idea of what happened there.

00:14:37.750 --> 00:14:39.490
Some data was collected.

00:14:39.490 --> 00:14:42.580
Some scientific
hypothesis was formulated.

00:14:42.580 --> 00:14:47.920
And then the data was here
to try to prove or disprove

00:14:47.920 --> 00:14:49.360
this scientific hypothesis.

00:14:49.360 --> 00:14:51.600
That's the usual
scientific process.

00:14:51.600 --> 00:14:55.300
And we need to understand how
the scientific process goes,

00:14:55.300 --> 00:14:58.030
because some of those things
might be actually questionable.

00:14:58.030 --> 00:15:02.140
Who is 100% sure that
study finds that students--

00:15:02.140 --> 00:15:04.600
do you think that you
benefit from waiting

00:15:04.600 --> 00:15:05.710
to declare a major?

00:15:05.710 --> 00:15:09.370
Right I would be
skeptical about this.

00:15:09.370 --> 00:15:13.360
I would be like, I don't want
to wait to declare a major.

00:15:13.360 --> 00:15:15.130
So what kind of
thing can we bring?

00:15:15.130 --> 00:15:17.170
Well maybe this
study studied people

00:15:17.170 --> 00:15:18.370
that were different from me.

00:15:18.370 --> 00:15:21.126
Or maybe the study
finds that this

00:15:21.126 --> 00:15:22.750
is beneficial for a
majority of people.

00:15:22.750 --> 00:15:23.890
I'm not a majority.

00:15:23.890 --> 00:15:24.724
I'm just one person.

00:15:24.724 --> 00:15:26.098
There's a bunch
of things that we

00:15:26.098 --> 00:15:28.540
need to understand what
those things actually mean.

00:15:28.540 --> 00:15:30.248
And we'll see that
those are actually not

00:15:30.248 --> 00:15:31.511
statements about individuals.

00:15:31.511 --> 00:15:33.760
They're not even statements
about the cohort of people

00:15:33.760 --> 00:15:35.050
they've actually looked at.

00:15:35.050 --> 00:15:37.270
They're statements
about a parameter

00:15:37.270 --> 00:15:40.860
of a distribution
that was used to model

00:15:40.860 --> 00:15:43.720
the benefit of waiting.

00:15:43.720 --> 00:15:45.040
So there's a lot of questions.

00:15:45.040 --> 00:15:46.840
And there are a lot of
layers that come into this.

00:15:46.840 --> 00:15:49.131
And we're going to want to
understand what was going on

00:15:49.131 --> 00:15:52.992
in there and try to peel it off
and understand what assumptions

00:15:52.992 --> 00:15:53.950
have been put in there.

00:15:53.950 --> 00:15:59.650
Even though it looks like
a totally legit study, out

00:15:59.650 --> 00:16:01.759
of those studies,
statistically, I

00:16:01.759 --> 00:16:04.050
think there's going to be
one that's going to be wrong.

00:16:04.050 --> 00:16:05.140
Well, maybe not one.

00:16:05.140 --> 00:16:07.930
But if I put a
long list of those,

00:16:07.930 --> 00:16:10.150
there would be a few that
would actually be wrong.

00:16:10.150 --> 00:16:12.840
If I put 20, there would
definitely be one that's wrong.

00:16:12.840 --> 00:16:13.840
So you have to see that.

00:16:13.840 --> 00:16:16.600
Every time you see 20 studies,
one is probably wrong.

00:16:16.600 --> 00:16:19.000
When there are studies
about drug effects,

00:16:19.000 --> 00:16:21.260
out of a list of 100,
one would be wrong.

00:16:21.260 --> 00:16:23.590
So we'll see what that means
and what I mean by that.

00:16:26.840 --> 00:16:30.980
Of course, not only studies
that make discoveries

00:16:30.980 --> 00:16:32.930
are actually making
the press titles.

00:16:32.930 --> 00:16:36.890
There's also the press
that talks about things

00:16:36.890 --> 00:16:38.480
that make no sense.

00:16:38.480 --> 00:16:41.600
I love this first experiment--
the salmon experiment.

00:16:41.600 --> 00:16:44.240
Actually, it was
a grad student who

00:16:44.240 --> 00:16:47.720
came to a neuroscience
poster session,

00:16:47.720 --> 00:16:50.780
pulled out this
poster, and explained

00:16:50.780 --> 00:16:53.210
the scientific experiment
that he was conducting,

00:16:53.210 --> 00:16:59.510
which consisted in taking a
previously frozen and thawed

00:16:59.510 --> 00:17:02.060
salmon, putting it
in an MRI, showing it

00:17:02.060 --> 00:17:07.760
pictures of violent images, and
recording its brain activity.

00:17:07.760 --> 00:17:11.780
And he was able to discover a
few voxels that were activated

00:17:11.780 --> 00:17:14.180
by those violent images.

00:17:14.180 --> 00:17:16.339
And can somebody tell
me what happened here?

00:17:19.200 --> 00:17:23.530
Was the salmon responding
to the violent activity?

00:17:23.530 --> 00:17:26.020
Basically, this is just
a statistical fluke.

00:17:26.020 --> 00:17:27.730
That's just randomness at play.

00:17:27.730 --> 00:17:29.936
There's so many voxels
that are recorded,

00:17:29.936 --> 00:17:31.310
and there's so
many fluctuations.

00:17:31.310 --> 00:17:32.630
There's always a
little bit of noise

00:17:32.630 --> 00:17:34.588
when you're in those
things, that some of them,

00:17:34.588 --> 00:17:36.472
just by chance, got lit up.

00:17:36.472 --> 00:17:38.680
And so we need to understand
how to correct for that.

00:17:38.680 --> 00:17:40.210
In this particular
instance, we need

00:17:40.210 --> 00:17:43.600
to have tools that tell us that,
well, finding three voxels that

00:17:43.600 --> 00:17:47.020
are activated for
that many voxels

00:17:47.020 --> 00:17:50.200
that you can find in
the salmon's brain

00:17:50.200 --> 00:17:53.030
is just too small of a number.

00:17:53.030 --> 00:17:56.050
Maybe we need to find a clump
of 20 of them, for example.

00:17:56.050 --> 00:17:57.670
All right, so
we're going to have

00:17:57.670 --> 00:18:02.710
mathematical tools that help us
find those particular numbers.

00:18:02.710 --> 00:18:07.780
I don't know if you ever
saw this one by John Oliver

00:18:07.780 --> 00:18:11.230
about phacking.

00:18:11.230 --> 00:18:14.110
Or actually, it said p-hacking.

00:18:14.110 --> 00:18:17.710
Basically, what John
Oliver is saying

00:18:17.710 --> 00:18:20.410
is actually a full-length-- like
there's long segments on this.

00:18:20.410 --> 00:18:24.220
And he was explaining how
there's a sociology question

00:18:24.220 --> 00:18:28.930
here about how there's a
huge incentive for scientists

00:18:28.930 --> 00:18:30.005
to publish results.

00:18:30.005 --> 00:18:31.630
You're not going to
say, you know what?

00:18:31.630 --> 00:18:33.400
This year, I found nothing.

00:18:33.400 --> 00:18:35.277
And so people are
trying to find things.

00:18:35.277 --> 00:18:36.860
And just by searching,
it's as if they

00:18:36.860 --> 00:18:39.070
were searching for all
the voxels in a brain

00:18:39.070 --> 00:18:41.920
until they find one that
was just lit up by chance.

00:18:41.920 --> 00:18:43.690
And so they just run
all these studies.

00:18:43.690 --> 00:18:46.960
And at some point, one will
be right just out of chance.

00:18:46.960 --> 00:18:49.630
And so we have to be very
careful about doing this.

00:18:49.630 --> 00:18:52.435
There's much more complicated
problems associated

00:18:52.435 --> 00:18:53.920
to what's called
p-hacking, which

00:18:53.920 --> 00:18:57.970
consists of violating the basic
assumptions, in particular,

00:18:57.970 --> 00:19:00.130
looking at the data,
and then formulating

00:19:00.130 --> 00:19:02.260
your scientific
assumption based on data,

00:19:02.260 --> 00:19:03.400
and then going back to it.

00:19:03.400 --> 00:19:04.358
Your idea doesn't work.

00:19:04.358 --> 00:19:05.810
Let's just formulate
another one.

00:19:05.810 --> 00:19:07.670
And if you are doing
this, all bets are off.

00:19:10.127 --> 00:19:11.710
The theory that we're
going to develop

00:19:11.710 --> 00:19:14.170
is actually for a very
clean use of data, which

00:19:14.170 --> 00:19:16.450
might be a little unpleasant.

00:19:16.450 --> 00:19:19.930
If you've had an army of
graduate students collecting

00:19:19.930 --> 00:19:21.730
genomic data for a
year, for example,

00:19:21.730 --> 00:19:23.920
maybe you don't
want to say, well,

00:19:23.920 --> 00:19:25.810
I had one hypothesis
that didn't work.

00:19:25.810 --> 00:19:27.940
Let's throw all the
data into the trash.

00:19:27.940 --> 00:19:30.650
And so we need to find
ways to be able to do this.

00:19:30.650 --> 00:19:34.750
And there's actually a
course been taught at BU.

00:19:34.750 --> 00:19:37.180
It's still in its early
stages, but something

00:19:37.180 --> 00:19:39.205
called "adaptive data
analysis" that will allow

00:19:39.205 --> 00:19:42.615
you to do these kind of things.

00:19:42.615 --> 00:19:43.569
Questions?

00:19:46.910 --> 00:19:49.870
OK, so of course,
statistics is not

00:19:49.870 --> 00:19:52.480
just for you to be
able to read the press.

00:19:52.480 --> 00:19:56.350
Statistics will probably
be used in whatever career

00:19:56.350 --> 00:19:58.570
path you choose for yourself.

00:19:58.570 --> 00:20:03.070
It started in the 10th century
in Netherlands for hydrology.

00:20:03.070 --> 00:20:06.410
Netherlands is basically
under water, under sea level.

00:20:06.410 --> 00:20:08.237
And so they wanted
to build some dikes.

00:20:08.237 --> 00:20:09.820
But once you're going
to build a dike,

00:20:09.820 --> 00:20:11.890
you want to make sure
that it's going to sustain

00:20:11.890 --> 00:20:13.875
some tides and some floods.

00:20:13.875 --> 00:20:15.250
And so in particular,
they wanted

00:20:15.250 --> 00:20:19.799
to build dikes that were high
enough, but not too high.

00:20:19.799 --> 00:20:21.340
You could always
say, well, I'm going

00:20:21.340 --> 00:20:25.750
to build a 500-meter dike,
and then I'm going to be safe.

00:20:25.750 --> 00:20:27.580
You want something
that's based on data.

00:20:27.580 --> 00:20:28.510
You want to make sure.

00:20:28.510 --> 00:20:30.400
And so in particular,
what did they do?

00:20:30.400 --> 00:20:33.700
Well, they collected
data for previous floods.

00:20:33.700 --> 00:20:36.220
And then they just
found a dike that

00:20:36.220 --> 00:20:37.720
was going to cover
all these things.

00:20:37.720 --> 00:20:40.060
Now, if you look at the
data they probably had,

00:20:40.060 --> 00:20:41.590
maybe it was scarce.

00:20:41.590 --> 00:20:43.484
Maybe they had 10 data points.

00:20:43.484 --> 00:20:44.900
And so for those
data points, then

00:20:44.900 --> 00:20:47.622
maybe they wanted to
sort of interpolate

00:20:47.622 --> 00:20:50.080
between those points, maybe
extrapolate for the larger one.

00:20:50.080 --> 00:20:51.810
Based on what they've
seen, maybe they

00:20:51.810 --> 00:20:53.740
have chances of
seeing something which

00:20:53.740 --> 00:20:56.530
is even larger than everything
they've seen before.

00:20:56.530 --> 00:21:00.710
And that's exactly the goal
of statistical modeling--

00:21:00.710 --> 00:21:04.030
being able to extrapolate
beyond the data that you have,

00:21:04.030 --> 00:21:08.050
guessing what you have
not seen yet might happen.

00:21:08.050 --> 00:21:10.390
When you buy insurance
for your car,

00:21:10.390 --> 00:21:14.050
or your apartment,
or your phone,

00:21:14.050 --> 00:21:16.240
there is a premium
that you have to pay.

00:21:16.240 --> 00:21:17.800
And this premium
has been determined

00:21:17.800 --> 00:21:21.370
based on how much you
are, in expectation, going

00:21:21.370 --> 00:21:22.860
to cost the insurance.

00:21:22.860 --> 00:21:25.870
It says, OK, this person
has, day a 10% chance

00:21:25.870 --> 00:21:28.200
of breaking their iPhone.

00:21:28.200 --> 00:21:30.047
An iPhone costs
that much to repair,

00:21:30.047 --> 00:21:31.630
so I'm going to
charge them that much.

00:21:31.630 --> 00:21:34.690
And then I'm going to add
an extra dollar for my time.

00:21:34.690 --> 00:21:36.850
That's basically how those
things are determined.

00:21:36.850 --> 00:21:39.220
And so this is using statistics.

00:21:39.220 --> 00:21:42.910
This is basically where
statistics is probably

00:21:42.910 --> 00:21:43.630
mostly used.

00:21:43.630 --> 00:21:45.610
I was personally
trained as an actuary.

00:21:45.610 --> 00:21:48.160
And that's me being a
statistician at an insurance

00:21:48.160 --> 00:21:50.270
company.

00:21:50.270 --> 00:21:55.460
Clinical trials-- this is also
one of the earliest success

00:21:55.460 --> 00:21:58.040
stories of statistics.

00:21:58.040 --> 00:21:59.570
It's actually now widespread.

00:21:59.570 --> 00:22:04.430
Every time a new drug is
approved for market by the FDA,

00:22:04.430 --> 00:22:08.570
it requires a very strict
regimen of testing with data,

00:22:08.570 --> 00:22:10.407
and control group,
and treatment group,

00:22:10.407 --> 00:22:11.990
and how many people
you need in there,

00:22:11.990 --> 00:22:17.390
and what kind of significance
you need for those things.

00:22:17.390 --> 00:22:19.405
In particular, those
things look like this,

00:22:19.405 --> 00:22:21.507
so now it's 5,000 patients.

00:22:21.507 --> 00:22:23.090
It depends on what
kind of drug it is,

00:22:23.090 --> 00:22:26.630
but for, say, 100
patients, 56 were cured,

00:22:26.630 --> 00:22:29.150
and 44 showed no improvement.

00:22:29.150 --> 00:22:31.310
Does the FDA consider that
this is a good number?

00:22:31.310 --> 00:22:37.970
Do they have a table for how
many patients were cured?

00:22:37.970 --> 00:22:39.380
Is there a placebo effect?

00:22:39.380 --> 00:22:41.060
Do I need a control
group of people that

00:22:41.060 --> 00:22:42.680
are actually getting a placebo?

00:22:42.680 --> 00:22:44.130
It's not clear,
all these things.

00:22:44.130 --> 00:22:46.462
And so there's a lot of
things to put into place.

00:22:46.462 --> 00:22:48.170
And there's a lot of
floating parameters.

00:22:48.170 --> 00:22:50.100
So hopefully, we're
going to be able to use

00:22:50.100 --> 00:22:52.040
statistical modeling
to shrink it down

00:22:52.040 --> 00:22:55.190
to a small number of
parameters to be able to ask

00:22:55.190 --> 00:22:56.480
very simple questions.

00:22:56.480 --> 00:22:59.630
"Is a drug effective" is
not a mathematical equation.

00:22:59.630 --> 00:23:02.420
But "Is p larger than 0.5?"

00:23:02.420 --> 00:23:04.411
is a mathematical
question And that's

00:23:04.411 --> 00:23:05.910
essentially we're
going to be doing.

00:23:05.910 --> 00:23:08.710
We're going to take this, is a
drug effective, to reducing to,

00:23:08.710 --> 00:23:12.460
is a variable larger than 0.5?

00:23:12.460 --> 00:23:15.040
Now, of course genetics
are using that.

00:23:15.040 --> 00:23:19.120
That's typically actually
the same size of data

00:23:19.120 --> 00:23:21.100
that you would
see for FMRI data.

00:23:21.100 --> 00:23:25.480
So this is actually
a study that I found.

00:23:25.480 --> 00:23:29.890
You have about 4,000 cases of
Alzheimer's and 8,000 control.

00:23:29.890 --> 00:23:32.700
So people without Alzheimer's--
that's what's called a control.

00:23:32.700 --> 00:23:34.480
That's something
just to make sure

00:23:34.480 --> 00:23:38.770
that you can see the
difference with people

00:23:38.770 --> 00:23:42.610
that are not affected by
either a drug or a disease.

00:23:42.610 --> 00:23:46.854
Is the gene APOE associated
with Alzheimer's disease?

00:23:46.854 --> 00:23:49.270
Everybody can see why this
would be an important question.

00:23:49.270 --> 00:23:50.228
We now have it crisper.

00:23:50.228 --> 00:23:52.570
It's targeted to
very specific genes.

00:23:52.570 --> 00:23:55.554
If we could edit it, or
knock it down, or knock it

00:23:55.554 --> 00:23:57.220
up, or boost it, maybe
we could actually

00:23:57.220 --> 00:23:58.178
have an impact on that.

00:23:58.178 --> 00:24:00.111
So those are very
important questions,

00:24:00.111 --> 00:24:02.360
because we have the technology
to target those things.

00:24:02.360 --> 00:24:04.630
But we need the answers
about what those things are.

00:24:04.630 --> 00:24:07.557
And there's a bunch
of other questions.

00:24:07.557 --> 00:24:09.890
The minute you're going to
talk to biologists about say,

00:24:09.890 --> 00:24:10.300
I can do that.

00:24:10.300 --> 00:24:11.390
They're going to
say, OK, are there

00:24:11.390 --> 00:24:12.880
any other genes
within the genes,

00:24:12.880 --> 00:24:15.046
or any particular snips
that I can actually look at?

00:24:15.046 --> 00:24:17.084
And they're looking at
very different questions.

00:24:17.084 --> 00:24:19.000
And when you start asking
all these questions,

00:24:19.000 --> 00:24:22.040
you have to be careful, because
you're reusing your data again.

00:24:22.040 --> 00:24:26.920
And it might lead you
to wrong conclusions.

00:24:26.920 --> 00:24:29.530
And those are all over
the place, those things.

00:24:29.530 --> 00:24:32.170
And that's why they go all
the way to John Oliver talking

00:24:32.170 --> 00:24:33.567
about them.

00:24:33.567 --> 00:24:35.025
Any questions about
those examples?

00:24:37.609 --> 00:24:38.900
So this is really a motivation.

00:24:38.900 --> 00:24:40.358
Again, we're not
going to just take

00:24:40.358 --> 00:24:46.350
this data set of those cases
and look at them in detail.

00:24:46.350 --> 00:24:49.370
So what is common to
all these examples?

00:24:49.370 --> 00:24:50.960
Like, why do we have
to use statistics

00:24:50.960 --> 00:24:53.120
for all those things?

00:24:53.120 --> 00:24:55.650
Well, there's the
randomness of the data.

00:24:55.650 --> 00:24:59.300
There's some effect that
we just don't understand--

00:24:59.300 --> 00:25:02.750
for example, the randomness
associated with the lining up

00:25:02.750 --> 00:25:04.010
of some voxels.

00:25:04.010 --> 00:25:06.650
Or the fact that as
far as the insurance

00:25:06.650 --> 00:25:09.050
is concerned whether you're
going to break your iPhone

00:25:09.050 --> 00:25:10.790
or not is essentially
a coin toss.

00:25:10.790 --> 00:25:11.810
Fully, it's biased.

00:25:11.810 --> 00:25:14.780
But it's a coin toss.

00:25:14.780 --> 00:25:16.770
From the perspective
of the statistician,

00:25:16.770 --> 00:25:18.590
those things are
actually random events.

00:25:18.590 --> 00:25:20.550
And we need to tame
this randomness,

00:25:20.550 --> 00:25:21.800
to understand this randomness.

00:25:21.800 --> 00:25:23.722
Is this going to be
a lot of randomness?

00:25:23.722 --> 00:25:25.430
Or is it going to be
a little randomness?

00:25:25.430 --> 00:25:26.846
Is it going to be
something that's

00:25:26.846 --> 00:25:29.715
like, out of their people--

00:25:32.990 --> 00:25:35.040
let's see, for example,
for the floods.

00:25:35.040 --> 00:25:38.000
Were the floods that I
saw consistently almost

00:25:38.000 --> 00:25:40.790
the same size?

00:25:40.790 --> 00:25:43.310
It was almost a rounding
error, or they're just

00:25:43.310 --> 00:25:44.366
really widespread.

00:25:44.366 --> 00:25:45.990
All these things, we
need to understand

00:25:45.990 --> 00:25:48.320
so we can understand
how to build those dikes

00:25:48.320 --> 00:25:54.390
or how to make decisions
based on those data.

00:25:54.390 --> 00:25:58.130
And we need to understand
this randomness.

00:25:58.130 --> 00:26:01.550
OK, so the associated
questions to randomness

00:26:01.550 --> 00:26:03.110
were actually
hidden in the text.

00:26:03.110 --> 00:26:05.680
So we talked about
the notion of average.

00:26:05.680 --> 00:26:08.672
Right, so as far as the
insurance is concerned,

00:26:08.672 --> 00:26:10.880
they want to know in average
with the probability is.

00:26:10.880 --> 00:26:13.970
Like, what is your chance of
actually breaking your iPhone?

00:26:13.970 --> 00:26:18.330
And that's what came in
this notion of fair premium.

00:26:18.330 --> 00:26:21.472
There's this notion
of quantifying chance.

00:26:21.472 --> 00:26:23.430
We don't want to talk
maybe only about average,

00:26:23.430 --> 00:26:26.280
maybe you want to cover say
99% percent of the floods.

00:26:26.280 --> 00:26:31.920
So we need to know what is
the height of a flood that's

00:26:31.920 --> 00:26:34.350
higher than 99% of the floods.

00:26:34.350 --> 00:26:36.960
But maybe there's 1%
of them, you know.

00:26:36.960 --> 00:26:38.790
When doomsday comes,
doomsday comes.

00:26:38.790 --> 00:26:40.470
Right, we're not
going to pay for it.

00:26:40.470 --> 00:26:43.187
All right, so that's
most of the floods.

00:26:43.187 --> 00:26:45.270
And then there's questions
of significance, right?

00:26:45.270 --> 00:26:47.880
So you know I give this
example, a second ago

00:26:47.880 --> 00:26:50.070
about clinical trials.

00:26:50.070 --> 00:26:51.180
I give you some numbers.

00:26:51.180 --> 00:26:55.350
Clearly the drug cured more
people than it did not.

00:26:55.350 --> 00:26:58.020
But does it mean that
it's significantly good,

00:26:58.020 --> 00:26:59.220
or was this just by chance.

00:26:59.220 --> 00:27:01.950
Maybe it's just that these
people just recovered.

00:27:01.950 --> 00:27:04.710
It's like you know
curing a common cold.

00:27:04.710 --> 00:27:06.840
And you feel like,
oh I got cured.

00:27:06.840 --> 00:27:09.540
But it's really you waited five
days and then you got cured.

00:27:09.540 --> 00:27:11.910
All right, so there's this
notion of significance,

00:27:11.910 --> 00:27:12.780
of variability.

00:27:12.780 --> 00:27:15.150
All these things
are actually notions

00:27:15.150 --> 00:27:18.270
that describe randomness
and quantify randomness

00:27:18.270 --> 00:27:19.560
into simple things.

00:27:19.560 --> 00:27:21.630
Randomness is a very
complicated beast.

00:27:21.630 --> 00:27:24.390
But we can summarize it into
things that we understand.

00:27:24.390 --> 00:27:27.630
Just like I am a
complicated object.

00:27:27.630 --> 00:27:29.610
I'm made of molecules,
and made of genes,

00:27:29.610 --> 00:27:31.290
and made of very
complicated things.

00:27:31.290 --> 00:27:34.890
But I can be summarized as
my name, my email address,

00:27:34.890 --> 00:27:37.270
my height and my weight,
and maybe for most of you,

00:27:37.270 --> 00:27:39.100
this is basically enough.

00:27:39.100 --> 00:27:41.220
You will recognize
me without having

00:27:41.220 --> 00:27:45.240
to do a biopsy on me
every time you see me.

00:27:45.240 --> 00:27:49.020
All right, so, to
understand randomness

00:27:49.020 --> 00:27:51.450
you have to go
through probability.

00:27:51.450 --> 00:27:53.370
Probability is the
study of randomness.

00:27:53.370 --> 00:27:54.130
That's what it is.

00:27:54.130 --> 00:27:57.750
That's what the first sentence
that a lecturer in probability

00:27:57.750 --> 00:27:58.890
will say.

00:27:58.890 --> 00:28:02.040
And so that's why I need the
pre-requisite, because this

00:28:02.040 --> 00:28:04.830
is what we're going to use
to describe the randomness.

00:28:04.830 --> 00:28:07.830
We'll see in a second how it
interacts with statistics.

00:28:07.830 --> 00:28:10.590
So sometimes, and actually
probably most of the time

00:28:10.590 --> 00:28:13.460
throughout your
semester on probability,

00:28:13.460 --> 00:28:15.810
randomness was very
well understood.

00:28:15.810 --> 00:28:18.420
When you saw a
probability problem, here

00:28:18.420 --> 00:28:19.860
was the chance of
this happening,

00:28:19.860 --> 00:28:21.720
here was the chance
of that happening.

00:28:21.720 --> 00:28:23.820
Maybe you had more
complicated questions

00:28:23.820 --> 00:28:26.720
that you had some basic
elements to answer.

00:28:26.720 --> 00:28:32.127
For example, the probability
that I have HBO is this much.

00:28:32.127 --> 00:28:34.710
And the probability that I watch
Game of Thrones is that much.

00:28:34.710 --> 00:28:38.160
And given that I play basketball
what is the probability--

00:28:38.160 --> 00:28:39.870
you had all these
crazy questions,

00:28:39.870 --> 00:28:42.360
but you were able to build them.

00:28:42.360 --> 00:28:45.340
But all the basic numbers
were given to you.

00:28:45.340 --> 00:28:48.240
Statistics will be about
finding those basic numbers.

00:28:48.240 --> 00:28:51.940
All right so some examples
that you've probably seen

00:28:51.940 --> 00:28:55.950
were dice, cards,
roulette, flipping coins.

00:28:55.950 --> 00:28:57.700
All of these things
are things that you've

00:28:57.700 --> 00:28:59.060
seen in a probability class.

00:28:59.060 --> 00:29:00.726
And the reason is
because it's very easy

00:29:00.726 --> 00:29:02.680
to describe the probability
of each outcome.

00:29:02.680 --> 00:29:05.620
For a die we know that
each face is going

00:29:05.620 --> 00:29:07.150
to come with probably 1/6.

00:29:07.150 --> 00:29:09.910
Now I'm not going to go into
a debate of whether this

00:29:09.910 --> 00:29:12.160
is pure randomness or
this is determinism.

00:29:12.160 --> 00:29:14.940
I think as a model
for actual randomness

00:29:14.940 --> 00:29:18.220
a die is a pretty good
number, flipping a coin

00:29:18.220 --> 00:29:20.170
is a pretty good model.

00:29:20.170 --> 00:29:22.600
So those are actually
a good thing.

00:29:22.600 --> 00:29:24.650
So the questions that you
would see, for example,

00:29:24.650 --> 00:29:26.320
in probabilities
are the following.

00:29:26.320 --> 00:29:27.640
I roll one die.

00:29:27.640 --> 00:29:31.300
Alice gets $1 if the number
of dots is less than three.

00:29:31.300 --> 00:29:35.230
Bob gets $2 if the number
of dots is less than two.

00:29:35.230 --> 00:29:37.510
Do you want to be Alice or
Bob given that your role is

00:29:37.510 --> 00:29:40.965
actually to make money.

00:29:40.965 --> 00:29:43.910
Yeah, you want to be Bob, right?

00:29:43.910 --> 00:29:45.410
So let's see why.

00:29:45.410 --> 00:29:47.300
So if you look at the
expectation of what

00:29:47.300 --> 00:29:48.500
Alice makes.

00:29:48.500 --> 00:29:51.140
So let's call it a.

00:29:51.140 --> 00:29:56.080
This is $1, with
probability 1/2.

00:29:56.080 --> 00:29:59.630
So 3/6, that's 1/2.

00:29:59.630 --> 00:30:02.820
And the expectation
of what Bob makes,

00:30:02.820 --> 00:30:11.370
this is $2 with probably
2/6 and that's 2/3.

00:30:11.370 --> 00:30:13.230
Which is definitely
larger than 1/2.

00:30:13.230 --> 00:30:17.109
So Bob's expectations
actually a bit higher.

00:30:17.109 --> 00:30:18.900
So those are the kind
of questions that you

00:30:18.900 --> 00:30:19.941
may ask with probability.

00:30:19.941 --> 00:30:21.930
I described to you
exactly, you use the fact

00:30:21.930 --> 00:30:25.337
that the die would get
less than three dots,

00:30:25.337 --> 00:30:26.420
with probability one half.

00:30:26.420 --> 00:30:27.090
We knew that.

00:30:27.090 --> 00:30:29.880
And I didn't have to describe
to you what was going on there.

00:30:29.880 --> 00:30:32.460
You didn't have to
collect data about a die.

00:30:32.460 --> 00:30:34.620
Same thing, you roll two dice.

00:30:34.620 --> 00:30:36.180
You choose a number
between 2 and 12

00:30:36.180 --> 00:30:42.100
and you win $100 if you choose
the sum of the two dice.

00:30:42.100 --> 00:30:45.052
Which number do you pick?

00:30:45.052 --> 00:30:46.812
What?

00:30:46.812 --> 00:30:47.312
AUDIENCE: 7.

00:30:47.312 --> 00:30:48.186
PHILIPPE RIGOLLET: 7.

00:30:48.186 --> 00:30:48.800
Why 7?

00:30:48.800 --> 00:30:50.430
AUDIENCE: It's the most likely.

00:30:50.430 --> 00:30:52.638
PHILIPPE RIGOLLET: That's
the most likely one, right?

00:30:52.638 --> 00:30:56.800
So your gain here will be
$100 times the probability

00:30:56.800 --> 00:31:00.220
that the sum of the two
dice, let's say x plus y,

00:31:00.220 --> 00:31:03.430
is equal to your little
z where a little z is

00:31:03.430 --> 00:31:04.478
the number you pick.

00:31:07.830 --> 00:31:10.350
So 7 is the most
likely to happen

00:31:10.350 --> 00:31:14.580
and that's the one that
maximizes this function of z.

00:31:14.580 --> 00:31:17.112
And for this you need to study
a more complicated function.

00:31:17.112 --> 00:31:18.820
But it's a function
that enables two die.

00:31:18.820 --> 00:31:21.720
But you can compute the
probability that x plus y

00:31:21.720 --> 00:31:26.490
is equal to z, for every
z between 2 and 12.

00:31:26.490 --> 00:31:29.010
So you know exactly what
the probabilities are

00:31:29.010 --> 00:31:30.930
and that's how you
start probability.

00:31:35.450 --> 00:31:38.660
So here that's
exactly what I said.

00:31:38.660 --> 00:31:43.550
You have a very simple process
that describes basic events.

00:31:43.550 --> 00:31:45.074
Probability 1/6
for each of them.

00:31:45.074 --> 00:31:46.490
And then you can
build up on that,

00:31:46.490 --> 00:31:49.120
and understand probably of
more complicated events.

00:31:49.120 --> 00:31:50.750
You can throw some
money in there.

00:31:50.750 --> 00:31:52.780
You can be build functions.

00:31:52.780 --> 00:31:56.960
You can do very complicated
things building on that.

00:31:56.960 --> 00:31:59.780
Now if I was a
statistician, a statistician

00:31:59.780 --> 00:32:01.640
would be the guy who
just arrived on earth,

00:32:01.640 --> 00:32:03.473
had never seen a die
and needs to understand

00:32:03.473 --> 00:32:05.840
that a die come up with
probably 1/6 on each side.

00:32:05.840 --> 00:32:08.030
And the way he would do
it is just to roll the die

00:32:08.030 --> 00:32:12.440
until he get some counts
and tries to estimate those.

00:32:12.440 --> 00:32:14.060
And maybe that guy
would come and say,

00:32:14.060 --> 00:32:16.440
well, you know,
actually, the probability

00:32:16.440 --> 00:32:23.500
that I get a 1 is 1/6 plus
0.001 and the probability

00:32:23.500 --> 00:32:27.650
that I get a 2 is
1/6 minus 0.005.

00:32:27.650 --> 00:32:29.910
And there would be some
fluctuations around this.

00:32:29.910 --> 00:32:31.868
And it's going to be his
role as a statistician

00:32:31.868 --> 00:32:33.500
to say, listen, this
is too complicated

00:32:33.500 --> 00:32:34.880
of a model for this thing.

00:32:34.880 --> 00:32:36.694
And these should all
be the same numbers.

00:32:36.694 --> 00:32:39.110
Just looking at data, they
should be all the same numbers.

00:32:39.110 --> 00:32:40.150
And that's part of the modeling.

00:32:40.150 --> 00:32:41.930
You make some
simplifying assumptions

00:32:41.930 --> 00:32:46.590
that essentially make your
questions more accurate.

00:32:46.590 --> 00:32:48.230
Now, of course, if
your model is wrong,

00:32:48.230 --> 00:32:50.120
if it's not true that
all the faces arrive

00:32:50.120 --> 00:32:54.410
with the same probability, then
you have a model error here.

00:32:54.410 --> 00:32:56.146
So we will be
making model errors.

00:32:56.146 --> 00:32:57.770
But that's going to
be the price to pay

00:32:57.770 --> 00:32:59.645
to be able to extract
anything from our data.

00:33:02.860 --> 00:33:07.299
So for more
complicated processes,

00:33:07.299 --> 00:33:09.840
so of course nobody's going to
waste their time rolling dice.

00:33:09.840 --> 00:33:11.370
I mean, I'm sure
you might have done

00:33:11.370 --> 00:33:13.860
this in AP stat or something.

00:33:13.860 --> 00:33:18.070
But the need is to estimate
parameters from data.

00:33:18.070 --> 00:33:19.830
All right, so for more
complicated things

00:33:19.830 --> 00:33:27.350
you might want to estimate
some density parameter

00:33:27.350 --> 00:33:29.400
on a particular set of material.

00:33:29.400 --> 00:33:31.620
And for this maybe you need
to beam something to it,

00:33:31.620 --> 00:33:33.449
and measure how fast
it's coming back.

00:33:33.449 --> 00:33:35.490
And you're going to have
some measurement errors.

00:33:35.490 --> 00:33:37.281
And maybe you need to
do that several times

00:33:37.281 --> 00:33:39.750
and you have a model for
the physical process that's

00:33:39.750 --> 00:33:40.960
actually going on.

00:33:40.960 --> 00:33:42.750
And physics is usually
a very good way

00:33:42.750 --> 00:33:46.560
to get models for
engineering perspective.

00:33:46.560 --> 00:33:49.320
But there's models
for sociology where we

00:33:49.320 --> 00:33:52.260
have no physical system, right.

00:33:52.260 --> 00:33:53.670
God knows how people interact.

00:33:53.670 --> 00:33:55.650
And maybe I'm going
to say that the way

00:33:55.650 --> 00:33:59.870
I make friends is by first
flipping a coin in my pocket.

00:33:59.870 --> 00:34:01.560
And with probability
2/3, I'm going

00:34:01.560 --> 00:34:02.730
to make my friend at work.

00:34:02.730 --> 00:34:04.410
And with probability
1/3 I'm going

00:34:04.410 --> 00:34:05.850
to make my friend at soccer.

00:34:05.850 --> 00:34:07.850
And once I make my
friends at soccer--

00:34:07.850 --> 00:34:09.750
I decide to make
my friend soccer.

00:34:09.750 --> 00:34:11.400
Then I will face
someone who's flipping

00:34:11.400 --> 00:34:14.780
the same coin with maybe be
slightly different parameters.

00:34:14.780 --> 00:34:16.230
But those things actually exist.

00:34:16.230 --> 00:34:18.750
There's models about how
friendships are formed.

00:34:18.750 --> 00:34:22.020
And the one I
described is called

00:34:22.020 --> 00:34:23.177
the mixed-membership model.

00:34:23.177 --> 00:34:25.260
So those are models that
are sort of hypothesized.

00:34:25.260 --> 00:34:29.250
And they're more reasonable
than taking into account

00:34:29.250 --> 00:34:31.650
all the things that made
you meet that person

00:34:31.650 --> 00:34:34.409
at that particular time.

00:34:34.409 --> 00:34:38.580
So the goal here--
so based on data now,

00:34:38.580 --> 00:34:41.370
once we have the model is going
to be reduced to maybe two,

00:34:41.370 --> 00:34:42.989
three, four
parameters, depending

00:34:42.989 --> 00:34:44.500
on how complex the model is.

00:34:44.500 --> 00:34:48.060
And then your goal will be
to estimate those parameters.

00:34:48.060 --> 00:34:51.610
So sometimes the randomness
we have here is real.

00:34:51.610 --> 00:34:56.940
So there's some true
randomness in some surveys.

00:34:56.940 --> 00:34:58.740
If I pick a random
student, as long

00:34:58.740 --> 00:35:00.870
as I believe that my random
number generator that

00:35:00.870 --> 00:35:04.200
will pick your random
ID is actually random,

00:35:04.200 --> 00:35:06.180
there is something
random about you.

00:35:06.180 --> 00:35:07.950
The student that
I pick at random

00:35:07.950 --> 00:35:09.390
will be a random student.

00:35:09.390 --> 00:35:13.540
The person that I call on
the phone is a random person.

00:35:13.540 --> 00:35:16.380
So there's some randomness
that I can build into my system

00:35:16.380 --> 00:35:20.310
by drawing something from
a random number generator.

00:35:20.310 --> 00:35:22.512
A biased coin is a random thing.

00:35:22.512 --> 00:35:24.220
It's not a very
interesting random thing.

00:35:24.220 --> 00:35:26.100
But it is a random thing.

00:35:26.100 --> 00:35:29.040
Again, if I wash out the
fact that it actually

00:35:29.040 --> 00:35:30.510
is a deterministic mechanism.

00:35:30.510 --> 00:35:33.230
But at a certain accuracy,
a certain granularity,

00:35:33.230 --> 00:35:36.090
this can be thought of as
a truly random experiment.

00:35:36.090 --> 00:35:39.180
Measurement error for example,
if you by some measurement

00:35:39.180 --> 00:35:39.850
device.

00:35:39.850 --> 00:35:42.760
or some optics
device, for example.

00:35:42.760 --> 00:35:45.020
You will have like standard
deviation and things that

00:35:45.020 --> 00:35:46.080
come on the side of the box.

00:35:46.080 --> 00:35:48.621
And it tells you, this will be
making some measurement error.

00:35:48.621 --> 00:35:51.480
And it's usually thermal noise
maybe, or things like this.

00:35:51.480 --> 00:35:54.570
And those are very
accurately described

00:35:54.570 --> 00:35:56.430
by some random phenomenon.

00:35:56.430 --> 00:36:01.260
But sometimes, and I'd say most
times, there's no randomness.

00:36:01.260 --> 00:36:02.520
There's no randomness.

00:36:02.520 --> 00:36:06.810
It's not like you breaking
your iPhone is a random event.

00:36:06.810 --> 00:36:09.390
This is just something
that we sweep--

00:36:09.390 --> 00:36:11.940
randomness is a big rug
under which we sweep

00:36:11.940 --> 00:36:13.650
everything we don't understand.

00:36:13.650 --> 00:36:15.900
And we just hope
that in average we've

00:36:15.900 --> 00:36:18.911
captured, the average
effect of what's going on.

00:36:18.911 --> 00:36:20.910
And the rest of it might
fluctuate to the right,

00:36:20.910 --> 00:36:22.470
might fluctuate to the left.

00:36:22.470 --> 00:36:26.010
But what remains is
just sort of randomness

00:36:26.010 --> 00:36:27.660
that can be averaged out.

00:36:27.660 --> 00:36:31.170
So, of course, this is
where the leap of faith is.

00:36:31.170 --> 00:36:33.480
We do not know whether we
were correct of doing this.

00:36:33.480 --> 00:36:35.957
Maybe we make some
huge systematic biases

00:36:35.957 --> 00:36:36.540
by doing this.

00:36:36.540 --> 00:36:39.330
Maybe we forget a very
important component.

00:36:39.330 --> 00:36:42.450
Right, for example, if I have--

00:36:42.450 --> 00:36:45.816
I don't know, let's
think of something--

00:36:49.220 --> 00:36:51.080
a drug for breast cancer.

00:36:51.080 --> 00:36:52.880
All right, and I
throw out the fact

00:36:52.880 --> 00:36:55.640
that my patient is
either a man or woman.

00:36:55.640 --> 00:36:57.530
I'm going to have some
serious model biases.

00:36:57.530 --> 00:36:58.030
Right.

00:36:58.030 --> 00:37:00.645
So if I say I'm going to
collect a random and patient.

00:37:00.645 --> 00:37:02.270
And said I'm going
to start doing this.

00:37:02.270 --> 00:37:04.790
There's some information
that I really need, clearly,

00:37:04.790 --> 00:37:06.880
to build into my model.

00:37:06.880 --> 00:37:10.730
And so the model should be
complicated enough, but not too

00:37:10.730 --> 00:37:11.420
complicated.

00:37:11.420 --> 00:37:13.220
Right so it should take
into account things

00:37:13.220 --> 00:37:17.375
there will systematically
be important.

00:37:17.375 --> 00:37:19.700
So, in particular, the
simple rule of thumb

00:37:19.700 --> 00:37:24.620
is, when you have a
complicated process,

00:37:24.620 --> 00:37:26.780
you can think of it as
being a simple process

00:37:26.780 --> 00:37:28.205
and some random noise.

00:37:28.205 --> 00:37:30.320
Now, again, the random
noise is everything

00:37:30.320 --> 00:37:33.320
you don't understand about
the complicated process.

00:37:33.320 --> 00:37:37.810
And the simple process is
everything you actually do.

00:37:37.810 --> 00:37:40.600
So good modeling,
and this is not

00:37:40.600 --> 00:37:43.920
where we'll be
seeing in this class,

00:37:43.920 --> 00:37:46.890
consistent choosing
plausible simple models.

00:37:46.890 --> 00:37:50.241
And this requires a tremendous
amount of domain knowledge.

00:37:50.241 --> 00:37:52.240
And that's why we're not
doing it in this class.

00:37:52.240 --> 00:37:54.240
This is not something where I
can make a blanket statement

00:37:54.240 --> 00:37:55.620
about making good modeling.

00:37:55.620 --> 00:37:58.020
You need to know, if I
were a statistician working

00:37:58.020 --> 00:38:00.990
on a study, I would have to
grill the person in front

00:38:00.990 --> 00:38:04.840
of me, the expert, for two hours
to know, but how about this?

00:38:04.840 --> 00:38:05.970
How about that?

00:38:05.970 --> 00:38:06.990
How does this work?

00:38:06.990 --> 00:38:08.940
So it requires to
understand a lot of things.

00:38:08.940 --> 00:38:14.070
There's this famous statistician
to whom this sentence is

00:38:14.070 --> 00:38:16.640
attributed, and it's
probably not his then,

00:38:16.640 --> 00:38:21.191
but Tukey said that he
loves being a statistician,

00:38:21.191 --> 00:38:23.190
because you get to play
in everybody's backyard.

00:38:23.190 --> 00:38:25.440
Right, so you get to
go and see people.

00:38:25.440 --> 00:38:28.260
And you get to understand, at
least to a certain extent, what

00:38:28.260 --> 00:38:29.430
their problems are.

00:38:29.430 --> 00:38:31.110
Enough that you
can actually build

00:38:31.110 --> 00:38:33.400
a reasonable model for what
they're actually doing.

00:38:33.400 --> 00:38:34.740
So you get to do some sociology.

00:38:34.740 --> 00:38:35.865
You get to do some biology.

00:38:35.865 --> 00:38:37.167
You get to do some engineering.

00:38:37.167 --> 00:38:39.000
And you get to do a lot
of different things.

00:38:39.000 --> 00:38:40.625
Right, so he was
actually at some point

00:38:40.625 --> 00:38:46.180
predicting the
presidential election.

00:38:46.180 --> 00:38:48.720
So, you see, you get to do
a lot of different things.

00:38:48.720 --> 00:38:50.850
But it requires a lot
of time to understand

00:38:50.850 --> 00:38:52.230
what problem you're working on.

00:38:52.230 --> 00:38:54.480
And if you have a particular
application in mind

00:38:54.480 --> 00:38:56.644
you're the best person to
actually understand this.

00:38:56.644 --> 00:38:58.560
So I'm just going to
give you the basic tools.

00:39:07.660 --> 00:39:11.705
So this is the circle of trust.

00:39:11.705 --> 00:39:14.010
No, this is really
just a simple graphic

00:39:14.010 --> 00:39:15.840
that tells you what's going on.

00:39:15.840 --> 00:39:19.170
When you do probability,
you're given the truth.

00:39:19.170 --> 00:39:24.330
Somebody tells you what
die God is rolling.

00:39:24.330 --> 00:39:27.490
So you know exactly what the
parameters of the problems are.

00:39:27.490 --> 00:39:29.490
And what you're trying
to do is to describe what

00:39:29.490 --> 00:39:31.410
the outcomes are going to be.

00:39:31.410 --> 00:39:34.260
You can say, if you're
rolling a fair die,

00:39:34.260 --> 00:39:36.480
you're going to have 1/6
of the time in your data

00:39:36.480 --> 00:39:37.699
you're going to have one.

00:39:37.699 --> 00:39:39.740
1/6 of the time you're
going to have to have two.

00:39:39.740 --> 00:39:42.156
And so you can describe-- if
I told you what the truth is,

00:39:42.156 --> 00:39:44.340
you could actually
go into a computer,

00:39:44.340 --> 00:39:46.080
either generate some data.

00:39:46.080 --> 00:39:51.030
Or you could describe to me
some more macro properties

00:39:51.030 --> 00:39:52.350
of what the data would be like.

00:39:52.350 --> 00:39:54.600
Oh, I would see a
bunch of numbers

00:39:54.600 --> 00:39:57.150
that would be centered
around 35, if I

00:39:57.150 --> 00:40:00.182
drew from a Gaussian
distribution centered at 35.

00:40:00.182 --> 00:40:01.890
Right, you would know
this kind of thing.

00:40:01.890 --> 00:40:07.400
I would know that it's very
unlikely that if my Gaussian

00:40:07.400 --> 00:40:08.940
has standard deviation--

00:40:08.940 --> 00:40:13.830
is centered on 0, say,
with standard deviation 3.

00:40:13.830 --> 00:40:17.380
It's very unlikely that I will
see numbers below minus 10

00:40:17.380 --> 00:40:18.750
in above 10, right?

00:40:18.750 --> 00:40:21.060
You know this, that you
basically will not see them.

00:40:21.060 --> 00:40:25.310
So you know from the truth,
from the distribution

00:40:25.310 --> 00:40:27.810
of a random variable that does
not have mu or sigmas, really

00:40:27.810 --> 00:40:28.860
numbers there.

00:40:28.860 --> 00:40:31.980
You know what data,
you're going to be having.

00:40:31.980 --> 00:40:33.895
Statistics is about
going backwards.

00:40:33.895 --> 00:40:37.590
It's saying, if I have
some data, what was

00:40:37.590 --> 00:40:39.420
the truth that generated it.

00:40:39.420 --> 00:40:41.790
And since there are so
many possible truths,

00:40:41.790 --> 00:40:44.040
Modeling says you
have to pick one

00:40:44.040 --> 00:40:47.190
of the simpler possible truths,
so that you can average out.

00:40:47.190 --> 00:40:49.720
Statistics basically
means averaging.

00:40:49.720 --> 00:40:51.540
You're averaging when
you do statistics.

00:40:51.540 --> 00:40:54.150
And averaging
means that if I say

00:40:54.150 --> 00:40:56.790
that I received--
so if I collect

00:40:56.790 --> 00:40:58.230
all your GPAs, for example.

00:40:58.230 --> 00:41:01.860
And my model is that
the possible GPAs

00:41:01.860 --> 00:41:03.450
are any possible numbers.

00:41:03.450 --> 00:41:05.297
And anybody can have
any possible GPA.

00:41:05.297 --> 00:41:06.880
This is going to be
a serious problem.

00:41:06.880 --> 00:41:09.420
But if I can summarize
those GPAs into two numbers,

00:41:09.420 --> 00:41:11.100
say, mean and
standard deviation,

00:41:11.100 --> 00:41:13.020
than I have a pretty
good description of what

00:41:13.020 --> 00:41:15.360
is going on, rather
than having to have

00:41:15.360 --> 00:41:16.730
to predict the full list.

00:41:16.730 --> 00:41:18.880
Right, if I learn a full
list of GPAs and I say,

00:41:18.880 --> 00:41:20.282
well this was the distribution.

00:41:20.282 --> 00:41:22.740
Then it's not going to be of
any use for me to predict what

00:41:22.740 --> 00:41:25.232
the GPA would be, or some
random student walking in,

00:41:25.232 --> 00:41:26.190
or something like this.

00:41:30.210 --> 00:41:34.680
So just to finish my rant about
probability versus statistics,

00:41:34.680 --> 00:41:37.230
this is a question you would
see in a probability-- this

00:41:37.230 --> 00:41:40.080
is a probabilistic question, and
this is a statistical question.

00:41:40.080 --> 00:41:42.690
The probabilistic question
is, previous studies

00:41:42.690 --> 00:41:45.150
showed that the drug
was 80% effective.

00:41:45.150 --> 00:41:46.230
So you know that.

00:41:46.230 --> 00:41:48.340
This is the effectiveness
of the drug.

00:41:48.340 --> 00:41:49.480
It's given to you.

00:41:49.480 --> 00:41:51.480
This is how your problem starts.

00:41:51.480 --> 00:41:54.930
Then we can anticipate that,
for a study on 100 patients,

00:41:54.930 --> 00:41:57.030
in average, 80 be cured.

00:41:57.030 --> 00:42:00.660
And at least 65 will be
cured with 99% chances.

00:42:00.660 --> 00:42:03.140
So again these are not--

00:42:03.140 --> 00:42:05.994
I'm not predicting on 100
patients exactly the number

00:42:05.994 --> 00:42:07.410
of them they're
going to be cured.

00:42:07.410 --> 00:42:08.970
And the number of
them that are not.

00:42:08.970 --> 00:42:11.160
But I'm actually
sort of predicting

00:42:11.160 --> 00:42:13.140
what things are going
to look like on average,

00:42:13.140 --> 00:42:17.610
or some macro properties of what
my data sets will look like.

00:42:17.610 --> 00:42:19.590
So with 99 percent
chances, that means

00:42:19.590 --> 00:42:23.550
that in 99.99% of the
data sets you will

00:42:23.550 --> 00:42:25.770
draw from this particular draw.

00:42:25.770 --> 00:42:30.870
99.99% of the cohort of 100
patients to whom you administer

00:42:30.870 --> 00:42:34.650
this drug, I will be able
to conclude that at least 65

00:42:34.650 --> 00:42:41.010
of them will be cured, on 99.99%
percent of those data sets.

00:42:41.010 --> 00:42:42.630
So that's a pretty
accurate prediction

00:42:42.630 --> 00:42:45.060
of what's going to happen.

00:42:45.060 --> 00:42:46.260
Statistics is the opposite.

00:42:46.260 --> 00:42:49.620
It says, well, I just know
that 78 out of 100 were cured.

00:42:49.620 --> 00:42:50.890
I have only one data set.

00:42:50.890 --> 00:42:53.340
I cannot make predictions
for all data sets.

00:42:53.340 --> 00:42:57.199
But I can go back
to the probability,

00:42:57.199 --> 00:42:59.490
make some inference about
what my probability will look

00:42:59.490 --> 00:43:03.360
like, and then say, OK, then
I can make those predictions

00:43:03.360 --> 00:43:04.150
later on.

00:43:04.150 --> 00:43:08.580
So when I start with
78/100 then maybe

00:43:08.580 --> 00:43:11.100
I'm actually, in this
case, I just don't know.

00:43:11.100 --> 00:43:16.410
My best guess here is
that I'm confident I

00:43:16.410 --> 00:43:19.800
have to add the extra error that
I bet you making by predicting

00:43:19.800 --> 00:43:25.650
that here, the drug is not 80%
effective but 78% effective.

00:43:25.650 --> 00:43:27.570
And they need some
error bars around this,

00:43:27.570 --> 00:43:30.960
that will hopefully contain 80%,
and then based on those error

00:43:30.960 --> 00:43:34.460
bars I'm going to make slightly
less precise predictions

00:43:34.460 --> 00:43:35.456
for the future.

00:43:39.440 --> 00:43:44.510
So, to conclude, so this
was, why statistics?

00:43:44.510 --> 00:43:46.530
So what is this course about?

00:43:46.530 --> 00:43:48.230
It's about understanding
the mathematics

00:43:48.230 --> 00:43:50.230
behind statistical methods.

00:43:50.230 --> 00:43:51.590
It's more of a tool.

00:43:51.590 --> 00:43:54.680
We're not going to have fun and
talk about algebraic geometry

00:43:54.680 --> 00:43:57.440
just for fun in
the middle of it.

00:43:57.440 --> 00:44:01.610
So it justifies quantitative
statements given some modeling

00:44:01.610 --> 00:44:03.500
assumptions, that we
will, in this class,

00:44:03.500 --> 00:44:06.730
mostly admit that the modeling
assumptions are correct.

00:44:06.730 --> 00:44:08.480
| the first part--
in this introduction,

00:44:08.480 --> 00:44:10.442
we will go through
them because it's

00:44:10.442 --> 00:44:12.650
very easy to forget what
the assumptions are actually

00:44:12.650 --> 00:44:13.310
making.

00:44:13.310 --> 00:44:15.650
But this will be a
pretty standard thing.

00:44:15.650 --> 00:44:18.680
The words you will
hear a lot are IID--

00:44:18.680 --> 00:44:20.630
independent and
identically distributed--

00:44:20.630 --> 00:44:23.120
that means that your data
is basically all the sams.

00:44:23.120 --> 00:44:28.070
And one data point is not
impacting another data point.

00:44:28.070 --> 00:44:30.320
Hopefully we can describe
some interesting mathematics

00:44:30.320 --> 00:44:31.400
arising in statistics.

00:44:31.400 --> 00:44:33.500
You know, if you've
taken linear algebra,

00:44:33.500 --> 00:44:36.590
maybe we can explain to you why.

00:44:36.590 --> 00:44:38.630
If you've done some
calculus, maybe we

00:44:38.630 --> 00:44:40.040
can do some
interesting calculus.

00:44:40.040 --> 00:44:42.980
We'll see how in the
spirit of applied math

00:44:42.980 --> 00:44:45.630
those things answer
interesting questions.

00:44:45.630 --> 00:44:49.550
And basically we'll try to
carve out a math toolbox that's

00:44:49.550 --> 00:44:52.300
useful for us statistics.

00:44:52.300 --> 00:44:55.280
And maybe you can extend it
to more sophisticated methods

00:44:55.280 --> 00:44:57.080
that we did not
cover in this class.

00:44:57.080 --> 00:44:59.180
In particular in the
immersion learning class,

00:44:59.180 --> 00:45:02.120
hopefully you'll be able to
have some statistical intuition

00:45:02.120 --> 00:45:04.590
about what is going on.

00:45:04.590 --> 00:45:06.680
So what this course
is not about,

00:45:06.680 --> 00:45:09.950
it's not about spending a lot
of time looking at data sets,

00:45:09.950 --> 00:45:13.310
and trying to understand
some statistical thinking

00:45:13.310 --> 00:45:14.330
kind of questions.

00:45:14.330 --> 00:45:16.550
So this is more of an applied
statistical perspective

00:45:16.550 --> 00:45:19.760
on things, or more modeling.

00:45:19.760 --> 00:45:22.150
So I'm going to typically
give you the model.

00:45:22.150 --> 00:45:23.510
And say this is a model.

00:45:23.510 --> 00:45:26.480
And this is how we're
going to build an estimator

00:45:26.480 --> 00:45:28.440
in the framework of this model.

00:45:28.440 --> 00:45:30.820
So for example, 18.075,
to a certain extent,

00:45:30.820 --> 00:45:32.945
is called "Statistical
Thinking and Data Analysis."

00:45:32.945 --> 00:45:36.500
So I'm hoping there is some
statistical thinking in there.

00:45:36.500 --> 00:45:38.840
We will not talk about
software implementation.

00:45:38.840 --> 00:45:42.270
Unfortunately, there's just
too little time in a semester.

00:45:42.270 --> 00:45:45.840
There's other courses that
are giving you some overview.

00:45:45.840 --> 00:45:49.355
So the main software
these days are R

00:45:49.355 --> 00:45:54.830
is the leading software I'd say
in statistics, both in academia

00:45:54.830 --> 00:45:58.100
and industry, lots of
packages, one every day

00:45:58.100 --> 00:46:00.140
that's probably coming out.

00:46:00.140 --> 00:46:03.410
But there's other things,
right, so now Python is probably

00:46:03.410 --> 00:46:09.110
catching up with all these
scikit-learn packages that

00:46:09.110 --> 00:46:10.130
are coming up.

00:46:10.130 --> 00:46:14.300
Julia has some
statistics in there,

00:46:14.300 --> 00:46:17.867
but it really if you were to
learn a statistical software,

00:46:17.867 --> 00:46:19.325
let's say you love
doing this, this

00:46:19.325 --> 00:46:21.590
would be the one that would
prove most useful for you

00:46:21.590 --> 00:46:22.190
in the future.

00:46:22.190 --> 00:46:26.700
It does not scale super well
to high dimensional data.

00:46:26.700 --> 00:46:28.550
So there is a class
an IDSS that actually

00:46:28.550 --> 00:46:31.080
uses R. It's called
IDS 0.12, I think

00:46:31.080 --> 00:46:36.520
it's called "Statistics,
Computation, and Applications,"

00:46:36.520 --> 00:46:37.850
or something like this.

00:46:37.850 --> 00:46:40.130
I'm also preparing,
with Peter Kempthorne,

00:46:40.130 --> 00:46:42.538
a course called
"Computational Statistics."

00:46:47.380 --> 00:46:50.830
It's going to be offered this
Spring as a special topics.

00:46:50.830 --> 00:46:55.880
And so Peter Kempthorne
will be teaching it.

00:46:55.880 --> 00:46:58.620
And this class
will actually focus

00:46:58.620 --> 00:47:00.807
on using R. And
even beyond that,

00:47:00.807 --> 00:47:02.390
it's not just going
to be about using.

00:47:02.390 --> 00:47:04.285
It's going to be
about understanding--

00:47:04.285 --> 00:47:05.910
just the same way we
we're going to see

00:47:05.910 --> 00:47:07.980
how math helps
you do statistics,

00:47:07.980 --> 00:47:09.870
it's going to help
see how math helps you

00:47:09.870 --> 00:47:12.531
do algorithims for statistics.

00:47:12.531 --> 00:47:15.030
All right, so we'll talk about
maximum likelihood estimator.

00:47:15.030 --> 00:47:16.910
Will need to maximize
some function.

00:47:16.910 --> 00:47:19.120
There's an optimization
toolbox to do that.

00:47:19.120 --> 00:47:20.970
And we'll see how we
can have specialized

00:47:20.970 --> 00:47:22.800
for statistics
for that, and what

00:47:22.800 --> 00:47:25.340
are the principles behind it.

00:47:25.340 --> 00:47:26.990
And you know, of
course, if you've

00:47:26.990 --> 00:47:29.230
taken AP stats you
probably think that stats

00:47:29.230 --> 00:47:31.250
is boring to death
because it was just

00:47:31.250 --> 00:47:34.310
a long laundry-list that
spent a lot of time on t-test.

00:47:34.310 --> 00:47:37.010
I'm pretty sure we're not going
to talk about t-test, well,

00:47:37.010 --> 00:47:38.480
maybe once.

00:47:38.480 --> 00:47:42.200
But this is not a matter of
saying you're going to do this.

00:47:42.200 --> 00:47:43.670
And this is a slight
variant of it.

00:47:43.670 --> 00:47:46.160
We're going to really try to
understand what's going on.

00:47:46.160 --> 00:47:49.820
So, admittedly, you have
not chosen the simplest way

00:47:49.820 --> 00:47:52.580
to get an A in
statistics on campus.

00:47:52.580 --> 00:47:54.470
All right, this is
not the easiest class.

00:47:54.470 --> 00:47:56.000
It might be
challenging at times,

00:47:56.000 --> 00:47:59.674
but I can promise you that
you will maybe suffer.

00:47:59.674 --> 00:48:01.340
But you will learn
something by the time

00:48:01.340 --> 00:48:02.381
you're out of this class.

00:48:02.381 --> 00:48:04.550
This will not be a
waste of your time.

00:48:04.550 --> 00:48:06.560
And you will be
able to understand,

00:48:06.560 --> 00:48:09.230
and not having to remember by
heart how those things actually

00:48:09.230 --> 00:48:10.680
work.

00:48:10.680 --> 00:48:13.750
Are there any questions?

00:48:13.750 --> 00:48:16.910
Anybody want to go to other
stats class on campus?

00:48:16.910 --> 00:48:18.860
Maybe it's not too late.

00:48:18.860 --> 00:48:21.360
OK.

00:48:21.360 --> 00:48:25.280
So let's do some statistics.

00:48:25.280 --> 00:48:29.600
So I see the time
now and it's 11:56,

00:48:29.600 --> 00:48:31.800
so we have another 30 minutes.

00:48:31.800 --> 00:48:35.880
I will typically
give you a three,

00:48:35.880 --> 00:48:37.820
four minute break if
you want to stretch,

00:48:37.820 --> 00:48:39.278
if you want to run
to the bathroom,

00:48:39.278 --> 00:48:45.220
if you want to check
your texts or Instagram.

00:48:45.220 --> 00:48:47.510
There was very little
content in this class,

00:48:47.510 --> 00:48:49.010
hopefully it was
entertaining enough

00:48:49.010 --> 00:48:51.170
that you don't need the break.

00:48:51.170 --> 00:48:55.510
But just in the future, so you
know you will have a break.

00:48:55.510 --> 00:49:01.670
So statistics, this is how it
starts, I'm French, what can

00:49:01.670 --> 00:49:05.750
I say I need to put
some French words.

00:49:05.750 --> 00:49:08.450
So this is not how office
hours are going to go down.

00:49:12.410 --> 00:49:16.370
Anybody know this sculpture
by a Rodin, The Kiss.

00:49:16.370 --> 00:49:18.520
Maybe probably The
Thinker is more famous.

00:49:18.520 --> 00:49:20.240
But this is actually
a pretty famous one.

00:49:20.240 --> 00:49:23.830
But is it really this
one, or is it this one.

00:49:23.830 --> 00:49:26.504
Anybody knows which one it is?

00:49:26.504 --> 00:49:27.910
This one?

00:49:27.910 --> 00:49:28.450
Or this one?

00:49:28.450 --> 00:49:30.628
AUDIENCE: The previous.

00:49:30.628 --> 00:49:32.532
PHILIPPE RIGOLLET: What's that?

00:49:32.532 --> 00:49:33.490
AUDIENCE: This one.

00:49:33.490 --> 00:49:33.990
PHILIPPE RIGOLLET:
It's this one.

00:49:33.990 --> 00:49:35.340
AUDIENCE: Final answer.

00:49:35.340 --> 00:49:39.240
PHILIPPE RIGOLLET: Yeah,
who votes for this one.

00:49:39.240 --> 00:49:40.380
OK.

00:49:40.380 --> 00:49:42.451
Who votes for that one?

00:49:42.451 --> 00:49:42.950
Thank you.

00:49:42.950 --> 00:49:45.770
I love that you do not want to
pronounce yourself with no data

00:49:45.770 --> 00:49:47.160
actually to make any decision.

00:49:47.160 --> 00:49:49.640
This is a total coin toss right.

00:49:49.640 --> 00:49:51.340
Turns out that there
is data, and there

00:49:51.340 --> 00:49:53.930
is in the very serious
journal Nature,

00:49:53.930 --> 00:49:56.210
someone published a
very serious paper which

00:49:56.210 --> 00:49:58.235
actually looks pretty serious.

00:49:58.235 --> 00:50:00.110
If you look at it, it's
like "Human Behavior:

00:50:00.110 --> 00:50:02.510
Adult persistence of
head-turning symmetry,"

00:50:02.510 --> 00:50:04.930
is a lot of fancy
words in there.

00:50:04.930 --> 00:50:07.310
And this, I'm not
kidding you, this study

00:50:07.310 --> 00:50:09.830
is about collecting
data of people kissing,

00:50:09.830 --> 00:50:12.080
and knowing if they bend
their head to the right

00:50:12.080 --> 00:50:14.130
or if they bend they
head to the left.

00:50:14.130 --> 00:50:15.630
And that's all it is.

00:50:15.630 --> 00:50:21.560
And so a neonatal
right-side preference

00:50:21.560 --> 00:50:25.510
makes a surprising romantic
reappearance in later life.

00:50:25.510 --> 00:50:27.830
There's an explanation for it.

00:50:27.830 --> 00:50:32.820
All right, so if we follow this
Nature which one is the one.

00:50:32.820 --> 00:50:33.540
This one?

00:50:33.540 --> 00:50:34.920
Or this one?

00:50:34.920 --> 00:50:35.980
This one, right?

00:50:35.980 --> 00:50:38.900
Head to the right.

00:50:38.900 --> 00:50:41.250
And to be fair, for
this class I was like,

00:50:41.250 --> 00:50:46.130
oh, I'm going to go and show
them what Google Images does.

00:50:46.130 --> 00:50:49.400
When you Google
kissing couple, it's

00:50:49.400 --> 00:50:51.740
inappropriate after
maybe the first picture.

00:50:51.740 --> 00:50:53.540
And so I cannot show you this.

00:50:53.540 --> 00:50:55.370
But you know you can
check for yourself.

00:50:55.370 --> 00:50:57.140
Though I would
argue, so this person

00:50:57.140 --> 00:51:00.890
here actually went
out in airports

00:51:00.890 --> 00:51:06.740
and took pictures of strangers
kissing and collecting data.

00:51:06.740 --> 00:51:10.970
And can somebody guess why
did he just not stay home

00:51:10.970 --> 00:51:12.875
and collect data
from Google Images

00:51:12.875 --> 00:51:17.300
by just googling
kissing couples.

00:51:17.300 --> 00:51:19.850
What's wrong with this data?

00:51:19.850 --> 00:51:22.754
I didn't know actually before I
actually went on Google Images.

00:51:22.754 --> 00:51:23.920
AUDIENCE: It can be altered?

00:51:23.920 --> 00:51:24.540
PHILIPPE RIGOLLET:
What was that?

00:51:24.540 --> 00:51:25.730
AUDIENCE: It can be altered.

00:51:25.730 --> 00:51:26.600
PHILIPPE RIGOLLET:
It can be altered.

00:51:26.600 --> 00:51:28.220
But, you know, who
would want to do this?

00:51:28.220 --> 00:51:29.420
I mean there's no
particular reason why

00:51:29.420 --> 00:51:31.920
you would want to flip an image
before putting it out there.

00:51:31.920 --> 00:51:34.610
I mean, you might, but
you know maybe they

00:51:34.610 --> 00:51:38.858
want to hide the brand of
your Gap shirt or something.

00:51:38.858 --> 00:51:42.260
AUDIENCE: I guess the people
who post pictures of themselves

00:51:42.260 --> 00:51:44.587
kissing on Google Images
are not representative

00:51:44.587 --> 00:51:45.670
of the general population.

00:51:45.670 --> 00:51:47.626
PHILIPPE RIGOLLET:
Yeah, that's very true.

00:51:47.626 --> 00:51:49.250
And actually it's
even worse than that.

00:51:49.250 --> 00:51:51.237
The people who post
pictures of themselves,

00:51:51.237 --> 00:51:52.820
are not posting
pictures of themselves

00:51:52.820 --> 00:51:54.290
or putting pictures
of the people

00:51:54.290 --> 00:51:55.760
that they took a picture of.

00:51:55.760 --> 00:51:59.367
And there usually is a
stock watermark on this.

00:51:59.367 --> 00:52:00.700
And it's basically stock images.

00:52:00.700 --> 00:52:03.440
Those are actors, and so
they've been directed to kiss

00:52:03.440 --> 00:52:06.060
and this is not a
natural thing to do.

00:52:06.060 --> 00:52:08.060
And actually, if you go
to Google Images-- and I

00:52:08.060 --> 00:52:10.602
encourage you to
do this, unless you

00:52:10.602 --> 00:52:12.310
don't want to see
inappropriate pictures,

00:52:12.310 --> 00:52:14.420
and they're mightily
inappropriate.

00:52:14.420 --> 00:52:19.520
And basically you will see
that this study is actually not

00:52:19.520 --> 00:52:20.310
working at all.

00:52:20.310 --> 00:52:21.470
I mean, I looked briefly.

00:52:21.470 --> 00:52:22.920
I didn't actually
collect numbers.

00:52:22.920 --> 00:52:26.460
But I didn't find a particular
tendency to bend right.

00:52:26.460 --> 00:52:28.770
If anything, it was actually
probably the opposite.

00:52:28.770 --> 00:52:31.380
And it's because those people
were directed to do it.

00:52:31.380 --> 00:52:34.975
They just don't actually
think about doing it.

00:52:34.975 --> 00:52:36.350
And also because
I think you need

00:52:36.350 --> 00:52:38.870
to justify writing in
your paper more than,

00:52:38.870 --> 00:52:41.720
I sat in front of my computer.

00:52:41.720 --> 00:52:46.820
So again, this
first sentence here,

00:52:46.820 --> 00:52:49.420
a neonatal right-side
preference--

00:52:49.420 --> 00:52:51.630
"is there a right
side preference?"

00:52:51.630 --> 00:52:53.770
is not a mathematical question.

00:52:53.770 --> 00:52:57.760
But we can start saying, let's
blah, and put some variables,

00:52:57.760 --> 00:52:59.990
and ask questions
about those variables.

00:52:59.990 --> 00:53:02.540
So you know x is actually
not a variable that's

00:53:02.540 --> 00:53:04.870
used very much in
statistics for parameters.

00:53:04.870 --> 00:53:07.129
But p is one, for parameter.

00:53:07.129 --> 00:53:09.420
And so you're going to take
your parameter of interest,

00:53:09.420 --> 00:53:12.100
p, As here is going to be
the proportion of couples.

00:53:12.100 --> 00:53:13.930
And that's among all couples.

00:53:13.930 --> 00:53:17.530
So here, if you talk about
statistical thinking,

00:53:17.530 --> 00:53:20.560
there would be a question about
what population this would

00:53:20.560 --> 00:53:22.080
actually be representative of.

00:53:22.080 --> 00:53:24.190
| usually this is
a call to your--

00:53:26.890 --> 00:53:30.280
sorry, I should not forget this
word it's important for you.

00:53:33.290 --> 00:53:34.720
OK, I forget this word.

00:53:34.720 --> 00:53:38.170
So this is--

00:53:38.170 --> 00:53:43.094
OK,

00:53:43.094 --> 00:53:44.510
So if you look at
this proportion,

00:53:44.510 --> 00:53:46.660
maybe these couples
that are in the study

00:53:46.660 --> 00:53:49.210
might be representative
only of couples in airports.

00:53:49.210 --> 00:53:51.700
Maybe they actually put on a
show for the other passengers.

00:53:51.700 --> 00:53:52.240
Who knows?

00:53:52.240 --> 00:53:54.342
You know, like, oh,
let's just do it as well.

00:53:54.342 --> 00:53:56.050
And just like the
people in Google Images

00:53:56.050 --> 00:53:57.175
they are actually doing it.

00:53:57.175 --> 00:53:58.930
So maybe you want
to just restrict it.

00:53:58.930 --> 00:54:01.120
But of course clearly if
it's appearing in Nature,

00:54:01.120 --> 00:54:04.420
it should not be only
about couples in airports.

00:54:04.420 --> 00:54:07.810
It's supposedly representative
of all couples in the world.

00:54:07.810 --> 00:54:10.210
And so here let's
just keep it vague,

00:54:10.210 --> 00:54:12.610
but you need to keep
in mind what population

00:54:12.610 --> 00:54:14.710
this is actually making
a statement about.

00:54:14.710 --> 00:54:20.120
So you have this full population
of people in the world.

00:54:20.120 --> 00:54:23.130
Right, so those are
all the couples.

00:54:23.130 --> 00:54:27.380
And this person went
ahead and collected data

00:54:27.380 --> 00:54:29.667
about a bunch of them.

00:54:29.667 --> 00:54:31.750
And we know that, in this
thing, there's basically

00:54:31.750 --> 00:54:33.521
a proportion of
them, that's like p,

00:54:33.521 --> 00:54:35.520
and that's the proportion
of them that's bending

00:54:35.520 --> 00:54:36.570
their head to the right.

00:54:36.570 --> 00:54:40.050
And so everybody on this side
is bending their heads right.

00:54:40.050 --> 00:54:41.550
And hopefully we
can actually sample

00:54:41.550 --> 00:54:42.716
this thing you're informing.

00:54:42.716 --> 00:54:44.640
That's basically the
process that's going on.

00:54:44.640 --> 00:54:47.250
So this is the
statistical experiment.

00:54:47.250 --> 00:54:49.589
We're going to observe
n kissing couples.

00:54:49.589 --> 00:54:51.880
So here we're going to put
as many variables as we can.

00:54:51.880 --> 00:54:53.505
So we don't have to
stick with numbers.

00:54:53.505 --> 00:54:55.630
And then we'll just
plug in the numbers.

00:54:55.630 --> 00:54:58.450
n kissing couples, and n
is also, in statistics,

00:54:58.450 --> 00:55:04.320
by the way, n is the size of
your sample 99.9% of the time.

00:55:04.320 --> 00:55:06.720
And collect the value
of each outcome.

00:55:06.720 --> 00:55:07.530
So we want numbers.

00:55:07.530 --> 00:55:08.696
We don't want right or left.

00:55:08.696 --> 00:55:12.300
So we're going to code them
by 0 and 1, pretty naturally.

00:55:12.300 --> 00:55:16.890
And then we're going to
estimate p which is unknown.

00:55:16.890 --> 00:55:18.229
So p is this area.

00:55:18.229 --> 00:55:19.770
And we're going to
estimate it simply

00:55:19.770 --> 00:55:24.570
by the proportion of right
So the proportion of crosses

00:55:24.570 --> 00:55:27.030
that actually fell
in the right side.

00:55:29.660 --> 00:55:33.340
So in this study
what you will find

00:55:33.340 --> 00:55:36.460
is that the numbers
that were collected

00:55:36.460 --> 00:55:43.690
were 124 couples, and that,
out of those 124, 80 of them

00:55:43.690 --> 00:55:46.570
turned their head to the right.

00:55:46.570 --> 00:55:49.240
So, p hat is a proportion.

00:55:49.240 --> 00:55:50.020
How do we do it?

00:55:50.020 --> 00:55:51.880
Well, you don't need
statistics for that.

00:55:51.880 --> 00:55:54.790
You're going to see
80 divided by 124.

00:55:54.790 --> 00:55:57.580
And you will find that
in this particular study

00:55:57.580 --> 00:56:00.139
64.5% of the
couples were bending

00:56:00.139 --> 00:56:01.180
their heads to the right.

00:56:01.180 --> 00:56:03.040
That's a pretty
large number, right?

00:56:03.040 --> 00:56:07.180
The question is if I picked
another 124 couples, maybe

00:56:07.180 --> 00:56:10.240
at different airports, different
times, would I see same number?

00:56:10.240 --> 00:56:11.920
Would this number be
all over the place?

00:56:11.920 --> 00:56:14.650
Would it be sometimes very
close to 120, or sometimes

00:56:14.650 --> 00:56:15.850
for close to 10?

00:56:15.850 --> 00:56:20.020
Or would it be-- is this number
actually fluctuating a lot.

00:56:20.020 --> 00:56:26.870
And so, hopefully not too much,
64.5 percent is definitely

00:56:26.870 --> 00:56:28.670
much larger than 50%.

00:56:28.670 --> 00:56:31.652
And so there seems to
be this preference.

00:56:31.652 --> 00:56:33.110
Now we're going to
have to quantify

00:56:33.110 --> 00:56:34.276
how much of this preference.

00:56:34.276 --> 00:56:38.240
Is this number significantly
larger than 50%?

00:56:38.240 --> 00:56:41.090
So if our data, for example,
was just three couples.

00:56:41.090 --> 00:56:43.370
I'm just going there,
I'm going to Logan.

00:56:43.370 --> 00:56:45.440
I call it, I do
right, left right.

00:56:45.440 --> 00:56:47.570
And then I see--

00:56:47.570 --> 00:56:53.060
see what's the name of
the fish place there?

00:56:53.060 --> 00:56:56.399
I go to I go to Wahlburgers
at Logan and I'm like,

00:56:56.399 --> 00:56:57.440
OK, I'm done for the day.

00:56:57.440 --> 00:56:58.610
I collect this data.

00:56:58.610 --> 00:57:02.120
I go home, and I'm like,
wow, 66.7% to the right.

00:57:02.120 --> 00:57:03.380
That's a pretty big number.

00:57:03.380 --> 00:57:06.650
It's even farther from
50% than this other guy.

00:57:06.650 --> 00:57:08.315
So I'm doing even better.

00:57:08.315 --> 00:57:10.190
But of course you know
that this is not true.

00:57:10.190 --> 00:57:12.237
Three people is definitely
not representative.

00:57:12.237 --> 00:57:13.820
If I stopped at the
first one, I would

00:57:13.820 --> 00:57:18.360
have actually-- at the first
two, I would have even 100%.

00:57:18.360 --> 00:57:21.290
So the question that statistics
is going to help us answer is,

00:57:21.290 --> 00:57:23.514
how large should the sample be?

00:57:23.514 --> 00:57:25.805
For some reason, I don't know
if you guys receive this,

00:57:25.805 --> 00:57:27.554
I'm an affiliate with
the Broad Institute,

00:57:27.554 --> 00:57:30.050
and since then I receive
one email per day

00:57:30.050 --> 00:57:32.240
that says, sample
size determination--

00:57:32.240 --> 00:57:33.950
how large should your sample be?

00:57:33.950 --> 00:57:36.040
Like, I know how large
should with my sample be.

00:57:36.040 --> 00:57:39.530
I've taken 18.650
multiple times.

00:57:39.530 --> 00:57:43.240
And so I know, but the
question is-- is 124

00:57:43.240 --> 00:57:45.270
a large enough number or not?

00:57:45.270 --> 00:57:47.540
Well, the answer is actually,
as usual, it depends.

00:57:47.540 --> 00:57:51.310
It will depend on the
true unknown value of p.

00:57:51.310 --> 00:57:56.590
But from those particular
values that we got, so 120 and--

00:57:56.590 --> 00:57:57.870
how many couples was there?

00:57:57.870 --> 00:57:58.750
80?

00:57:58.750 --> 00:58:02.240
We actually can
make some question.

00:58:02.240 --> 00:58:07.750
So here we said that
80 was larger than 50--

00:58:07.750 --> 00:58:12.310
was allowing us to
conclude at 64.5%.

00:58:12.310 --> 00:58:17.520
So it could be one reason to
say that it was larger than 50%.

00:58:17.520 --> 00:58:23.230
50% of 124 is 62.

00:58:23.230 --> 00:58:24.910
So the question is,
would I be would I

00:58:24.910 --> 00:58:28.750
be willing to make
this conclusion at 63?

00:58:28.750 --> 00:58:30.630
Is that a number that
would convince you?

00:58:30.630 --> 00:58:34.030
Who would be convinced by 63?

00:58:34.030 --> 00:58:35.465
who would be convinced by 72?

00:58:38.410 --> 00:58:40.404
Who would be convinced by 75?

00:58:40.404 --> 00:58:42.820
Hopefully the number of hands
that are raised should grow.

00:58:46.120 --> 00:58:48.820
Who would be convinced by 80?

00:58:48.820 --> 00:58:51.190
All right, so basically
those numbers actually

00:58:51.190 --> 00:58:52.720
don't come from anywhere.

00:58:52.720 --> 00:58:56.740
This 72 would be the number that
you would need for a study--

00:58:56.740 --> 00:58:58.662
most statistical studies
would be the number

00:58:58.662 --> 00:58:59.620
that they would retain.

00:58:59.620 --> 00:59:01.870
That's not for 124.

00:59:01.870 --> 00:59:04.360
You would need to see
72 that turn their head

00:59:04.360 --> 00:59:07.390
right to actually
make this conclusion.

00:59:07.390 --> 00:59:08.977
And then 75--

00:59:08.977 --> 00:59:11.560
So we'll see that there's many
ways to come to this conclusion

00:59:11.560 --> 00:59:12.934
because, as you
can see, this was

00:59:12.934 --> 00:59:15.250
published in Nature with 80.

00:59:15.250 --> 00:59:15.920
So that was OK.

00:59:15.920 --> 00:59:17.503
So 80 is actually a
very large number.

00:59:17.503 --> 00:59:20.116
This is 99 point--

00:59:20.116 --> 00:59:24.250
this 99% -- no, so
this is 95% confidence.

00:59:24.250 --> 00:59:26.290
This is 99% confidence.

00:59:26.290 --> 00:59:29.330
And this is 99.9%
percent confidence.

00:59:29.330 --> 00:59:34.210
So if you said 80 you're a
very conservative person.

00:59:34.210 --> 00:59:36.930
Starting at 72, you can
start making this conclusion.

00:59:39.820 --> 00:59:41.260
To understand
this, we need to do

00:59:41.260 --> 00:59:45.880
our little mathematical
kitchen here,

00:59:45.880 --> 00:59:49.810
and we need to do some modeling.

00:59:49.810 --> 00:59:51.460
So we need to
understand by modeling--

00:59:51.460 --> 00:59:55.000
we need understand what
random process we think

00:59:55.000 --> 00:59:57.190
this data is generating from.

00:59:57.190 --> 00:59:59.180
So it's going to have
some unknown parameters,

00:59:59.180 --> 01:00:00.190
unlike in probability.

01:00:00.190 --> 01:00:02.800
But we need to have just
basically everything written

01:00:02.800 --> 01:00:04.900
except for the values
of the parameters.

01:00:04.900 --> 01:00:08.530
When I said a die is coming
uniformly with probably 1/6

01:00:08.530 --> 01:00:12.220
then I need to have, say
maybe with probability-- maybe

01:00:12.220 --> 01:00:14.260
I should say here
are six numbers,

01:00:14.260 --> 01:00:18.550
and I need to just
fill those numbers.

01:00:18.550 --> 01:00:23.140
So for i equal 1 to
n, I'm going to define

01:00:23.140 --> 01:00:27.140
Ri to be the indicator.

01:00:27.140 --> 01:00:29.950
An indicator is just something
that takes value 1 if something

01:00:29.950 --> 01:00:31.330
is true, and 0 if not.

01:00:31.330 --> 01:00:34.450
So it's an indicator
that i-th couple

01:00:34.450 --> 01:00:36.820
turns the head to the right.

01:00:36.820 --> 01:00:39.580
So, Ri, so it's indexed by i.

01:00:39.580 --> 01:00:42.290
And it's one if the i-th couple
turns their head to the right,

01:00:42.290 --> 01:00:45.820
and 0 if it's--

01:00:45.820 --> 01:00:48.640
well actually, I guess they can
probably kiss straight, right?

01:00:48.640 --> 01:00:51.500
So that would be weird, but
they might be able to do this.

01:00:51.500 --> 01:00:54.280
So let's say not right.

01:00:54.280 --> 01:00:56.710
Then the estimator of
p, we said, was p hat.

01:00:56.710 --> 01:00:58.360
It was just the
ratio of two numbers.

01:00:58.360 --> 01:01:02.410
But really what it is is
I count, I sum those Ri's.

01:01:02.410 --> 01:01:07.360
Since I only add those that
take value 1, what this is is--

01:01:07.360 --> 01:01:10.330
this sum here is actually just
counting the number of 1's.

01:01:10.330 --> 01:01:13.810
Which is another way to say it's
counting the number of couples

01:01:13.810 --> 01:01:15.620
that are kissing to the right.

01:01:15.620 --> 01:01:18.520
And here I don't even
have to tell you anything

01:01:18.520 --> 01:01:20.230
about the numbers or anything.

01:01:20.230 --> 01:01:21.940
I can only keep track of--

01:01:21.940 --> 01:01:24.240
first couple is a 0
second couple is a 1,

01:01:24.240 --> 01:01:25.251
third couple is 0.

01:01:25.251 --> 01:01:27.250
The data set-- you can
actually find it online--

01:01:27.250 --> 01:01:29.646
is actually a sequence
of 0's and 1's.

01:01:29.646 --> 01:01:31.270
Now clearly for the
question that we're

01:01:31.270 --> 01:01:32.950
asking about this
proportion, I don't

01:01:32.950 --> 01:01:34.980
need to keep track of
all this information.

01:01:34.980 --> 01:01:37.150
All I need to keep
track of is the number

01:01:37.150 --> 01:01:39.130
of 0's and the number of 1's.

01:01:39.130 --> 01:01:42.340
Those are completely
interchangeable.

01:01:42.340 --> 01:01:44.440
There's no time effect in this.

01:01:44.440 --> 01:01:48.370
The first couple is no
different than the 15th couple.

01:01:48.370 --> 01:01:50.380
So we call this Rn bar.

01:01:50.380 --> 01:01:52.960
That's going to be a very
standard notation that we use.

01:01:52.960 --> 01:01:55.780
R might be replaced by
other letters like x--

01:01:55.780 --> 01:01:58.150
so xn bar, yn bar.

01:01:58.150 --> 01:02:00.700
And this thing
essentially means that I

01:02:00.700 --> 01:02:04.630
average the R's, or the
Ri's over n of them.

01:02:04.630 --> 01:02:06.100
And the bar means the average.

01:02:06.100 --> 01:02:09.460
So I divide by n the
total number of 1's.

01:02:09.460 --> 01:02:13.630
So here this sum was equal
to 80 in our example and n

01:02:13.630 --> 01:02:16.820
was equal to 124.

01:02:16.820 --> 01:02:18.370
Now this is an estimator.

01:02:18.370 --> 01:02:20.380
So an estimator is
different from an estimate.

01:02:20.380 --> 01:02:21.580
An estimate is a number.

01:02:21.580 --> 01:02:23.680
My estimate was 64.5.

01:02:23.680 --> 01:02:29.502
My estimator is this thing where
I keep all the variables free.

01:02:29.502 --> 01:02:31.210
And in particular, I
keep those variables

01:02:31.210 --> 01:02:34.720
to be random because I'm going
to think of a random couple

01:02:34.720 --> 01:02:37.360
kissing left to right as the
outcome of a random process,

01:02:37.360 --> 01:02:41.320
just like flipping a coin
be getting heads or tails.

01:02:41.320 --> 01:02:43.626
And so this thing here
is a random variable, Ri.

01:02:43.626 --> 01:02:46.250
And this average is, of course,
an average of random variables.

01:02:46.250 --> 01:02:47.500
It's itself a random variable.

01:02:47.500 --> 01:02:49.330
So an estimator is
a random variable.

01:02:49.330 --> 01:02:51.972
An estimate is the realization
of a random variable,

01:02:51.972 --> 01:02:53.680
or, in other words,
is the value that you

01:02:53.680 --> 01:02:56.140
get for this random variable
once you plug in the numbers

01:02:56.140 --> 01:02:58.960
that you've collected.

01:02:58.960 --> 01:03:01.030
So I can talk about the
accuracy of an estimator.

01:03:01.030 --> 01:03:02.470
Accuracy means what?

01:03:02.470 --> 01:03:04.950
Well, what would we
want for an estimator?

01:03:04.950 --> 01:03:06.892
Maybe we won't want it
to fluctuate too much.

01:03:06.892 --> 01:03:07.850
It's a random variable.

01:03:07.850 --> 01:03:11.050
So I'm talking about the
accuracy of a random variable.

01:03:11.050 --> 01:03:13.740
So maybe I don't want
it to be too volatile.

01:03:13.740 --> 01:03:16.440
I could have one
estimator which would be--

01:03:16.440 --> 01:03:20.010
just throw out 182
couples, keep only 2

01:03:20.010 --> 01:03:21.420
and average those two numbers.

01:03:21.420 --> 01:03:23.040
That's definitely
a worse estimator

01:03:23.040 --> 01:03:25.140
than keeping all of the 124.

01:03:25.140 --> 01:03:26.641
So I need to find
a way to say that.

01:03:26.641 --> 01:03:28.140
And what I'm going
to be able to say

01:03:28.140 --> 01:03:30.060
is that the number is
going to be fluctuating.

01:03:30.060 --> 01:03:31.669
If I take another
two couples, I'm

01:03:31.669 --> 01:03:33.210
going to be I'm
probably going to get

01:03:33.210 --> 01:03:34.590
a completely different number.

01:03:34.590 --> 01:03:38.400
But if I take another 124
couples two days later,

01:03:38.400 --> 01:03:40.260
maybe I'm going to have
a very number that's

01:03:40.260 --> 01:03:43.140
very close to 64.5%.

01:03:43.140 --> 01:03:43.969
So that's one way.

01:03:43.969 --> 01:03:46.260
The other thing we would like
about this estimator it's

01:03:46.260 --> 01:03:47.767
actually--

01:03:47.767 --> 01:03:49.350
maybe it's not too
volatile-- but also

01:03:49.350 --> 01:03:54.300
we want it to be close to the
number that we're looking for.

01:03:54.300 --> 01:03:55.410
Here is an estimator.

01:03:55.410 --> 01:03:57.150
It's a beautiful variable.

01:03:57.150 --> 01:04:00.330
72%, that's an estimator.

01:04:00.330 --> 01:04:02.520
Go out there just do
your favorite study

01:04:02.520 --> 01:04:06.310
about drug performance.

01:04:06.310 --> 01:04:10.440
And then they're going to
call you, MIT student taking

01:04:10.440 --> 01:04:12.011
statistics, they
say, so how are you

01:04:12.011 --> 01:04:13.260
going to build your estimator?

01:04:13.260 --> 01:04:15.450
We've collected those 5,000
or something like that.

01:04:15.450 --> 01:04:17.160
I'm just going to spit out 72%.

01:04:17.160 --> 01:04:19.200
Whatever the data says,
that's an estimator.

01:04:19.200 --> 01:04:21.510
It's a stupid estimator
but it is an estimator.

01:04:21.510 --> 01:04:23.360
But this is estimator
is very not volatile.

01:04:23.360 --> 01:04:25.390
Every time you're going
to have a new study,

01:04:25.390 --> 01:04:27.640
even if you change fields,
it's still going to be 72%.

01:04:27.640 --> 01:04:29.190
This is beautiful.

01:04:29.190 --> 01:04:31.590
And the problem is
that's probably not

01:04:31.590 --> 01:04:34.164
very close to the value you're
actually trying to estimate.

01:04:34.164 --> 01:04:35.080
So we need two things.

01:04:35.080 --> 01:04:36.996
We need are estimated
to be a random variable.

01:04:36.996 --> 01:04:39.240
So think in terms of densities.

01:04:39.240 --> 01:04:42.740
We want the density
to be pretty narrow.

01:04:42.740 --> 01:04:46.240
We want this thing
to have very little--

01:04:46.240 --> 01:04:52.570
so this is definitely
better than this.

01:04:52.570 --> 01:04:55.830
But also, we want the number
that we're interested in, p,

01:04:55.830 --> 01:04:57.475
to be very close to this--

01:04:57.475 --> 01:05:00.260
to be close to the values that
this thing is likely to take.

01:05:00.260 --> 01:05:04.020
If p is here, this is
not very good for us.

01:05:04.020 --> 01:05:06.872
So that's basically the things
we're going to be looking at.

01:05:06.872 --> 01:05:08.580
The first one is
referred to as variance.

01:05:08.580 --> 01:05:10.163
The second one is
referred to as bias.

01:05:10.163 --> 01:05:14.380
Those things come all
over in statistics.

01:05:14.380 --> 01:05:16.810
So we need to
understand a model.

01:05:16.810 --> 01:05:20.907
So here's the model that we have
for this particular problem.

01:05:20.907 --> 01:05:22.990
So we need to make assumptions
on the observations

01:05:22.990 --> 01:05:23.650
that we see.

01:05:23.650 --> 01:05:25.620
So we said we're going to assume
that the random variable--

01:05:25.620 --> 01:05:27.160
that's not too much
of a leap of faith.

01:05:27.160 --> 01:05:29.243
We're just sweeping under
the rug everything thing

01:05:29.243 --> 01:05:31.180
we don't understand
about those couples.

01:05:31.180 --> 01:05:33.130
And the assumption
that we make is

01:05:33.130 --> 01:05:36.000
that Ri is a random variable.

01:05:36.000 --> 01:05:38.120
This one you will
forget very soon.

01:05:38.120 --> 01:05:41.950
The second one is that
each of the Ri's is--

01:05:41.950 --> 01:05:45.310
so it's a random variable
that takes value 0 or 1.

01:05:45.310 --> 01:05:47.024
Anybody can suggest
the distribution

01:05:47.024 --> 01:05:48.065
for this random variable?

01:05:48.065 --> 01:05:49.363
AUDIENCE: Bernoulli.

01:05:49.363 --> 01:05:50.363
PHILIPPE RIGOLLET: What?

01:05:50.363 --> 01:05:51.500
AUDIENCE: Bernoulli.

01:05:51.500 --> 01:05:51.980
PHILIPPE RIGOLLET:
Bernoulli, right?

01:05:51.980 --> 01:05:53.210
And it's actually beautiful.

01:05:53.210 --> 01:05:56.770
This is where you have to do
the least statistical modeling.

01:05:56.770 --> 01:05:59.164
A random variable that
takes value 0 or 1

01:05:59.164 --> 01:06:00.080
is always a Bernoulli.

01:06:00.080 --> 01:06:02.581
That's the simplest variable
you can ever think of.

01:06:02.581 --> 01:06:04.580
Any variable that takes
only two possible values

01:06:04.580 --> 01:06:06.155
can be reduced to a Bernoulli.

01:06:06.155 --> 01:06:10.421
OK, so this is a Bernoulli.

01:06:10.421 --> 01:06:12.420
And here we make the
assumption that it actually

01:06:12.420 --> 01:06:16.140
takes parameter p.

01:06:16.140 --> 01:06:17.850
And there's an assumption here.

01:06:17.850 --> 01:06:21.427
Anybody can tell me
what the assumption is?

01:06:21.427 --> 01:06:22.780
AUDIENCE: It's the same.

01:06:22.780 --> 01:06:24.530
PHILIPPE RIGOLLET:
Yeah, it's same, right?

01:06:24.530 --> 01:06:26.194
I could have said
p i, but it's p.

01:06:26.194 --> 01:06:28.110
And that's where I'm
going to be able to start

01:06:28.110 --> 01:06:29.360
getting to do some statistics.

01:06:29.360 --> 01:06:31.930
It's that I'm going to start
to be able to pull information

01:06:31.930 --> 01:06:32.896
across all my guys.

01:06:32.896 --> 01:06:34.270
If I assume that
they're all pi's

01:06:34.270 --> 01:06:36.340
completely uncoupled
with each other.

01:06:36.340 --> 01:06:37.480
Then I'm in trouble.

01:06:37.480 --> 01:06:39.741
There's nothing I
can actually get.

01:06:39.741 --> 01:06:41.740
And then I'm going to
assume that those guys are

01:06:41.740 --> 01:06:42.940
mutually independent.

01:06:42.940 --> 01:06:45.530
And most of the time they
will just say independent.

01:06:45.530 --> 01:06:48.760
Meaning that, it's not like all
these guys called each other

01:06:48.760 --> 01:06:50.110
and it's actually a flash mob.

01:06:50.110 --> 01:06:53.200
And they were like, let's all
turn our left side to the left.

01:06:53.200 --> 01:06:54.790
And then this is
definitely not going

01:06:54.790 --> 01:06:59.910
to give you a valid conclusion.

01:06:59.910 --> 01:07:02.400
So, again. randomness is
a way of modeling lack

01:07:02.400 --> 01:07:03.210
of information.

01:07:03.210 --> 01:07:05.409
Here there is a way
to figure it out.

01:07:05.409 --> 01:07:07.200
Maybe I could have
followed all those guys,

01:07:07.200 --> 01:07:09.040
and knew exactly what
they were-- maybe

01:07:09.040 --> 01:07:11.670
I could have looked at
pictures of them in the womb

01:07:11.670 --> 01:07:14.354
and guess how they were
turning-- by the way that's

01:07:14.354 --> 01:07:16.020
one of the conclusions,
they're guessing

01:07:16.020 --> 01:07:17.436
that we turn our
head to the right

01:07:17.436 --> 01:07:21.180
because our head is turned
to the right in the womb.

01:07:21.180 --> 01:07:24.840
So we don't know what goes
on in the kissers minds.

01:07:24.840 --> 01:07:26.730
And there's, you know,
physics, sociology.

01:07:26.730 --> 01:07:28.521
There's a lot of things
that could help us,

01:07:28.521 --> 01:07:31.220
but it's just too
complicated to keep track of,

01:07:31.220 --> 01:07:34.550
or too expensive
for many instances

01:07:34.550 --> 01:07:37.310
Now again, the nicest
part of this modeling

01:07:37.310 --> 01:07:39.560
was the fact that Ri's
take only two values, which

01:07:39.560 --> 01:07:41.900
mean that this conclusion
that they were Bernoulli

01:07:41.900 --> 01:07:43.220
was totally free for us.

01:07:43.220 --> 01:07:45.530
Once we know it's a random
variable, it's a Bernoulli.

01:07:45.530 --> 01:07:47.720
Now they could have
been, as we said,

01:07:47.720 --> 01:07:51.910
they could have been a
Bernoulli with parameter p i.

01:07:51.910 --> 01:07:55.420
For each i, I could have
put a different parameter,

01:07:55.420 --> 01:07:57.672
but I just don't have
enough information.

01:07:57.672 --> 01:07:58.630
What would I have said?

01:07:58.630 --> 01:08:00.921
I would say, well the first
couple turned to the right.

01:08:00.921 --> 01:08:04.600
p1 has to be one,
that's my best guess.

01:08:04.600 --> 01:08:06.910
The second couple
kiss to the left,

01:08:06.910 --> 01:08:10.690
well, p2 should be 0,
that's my best guess.

01:08:10.690 --> 01:08:14.710
And so basically I
need to have to be

01:08:14.710 --> 01:08:16.240
able to average my information.

01:08:16.240 --> 01:08:19.649
And the way I do it is by
coupling all these guys,

01:08:19.649 --> 01:08:22.490
pi's to be the same p for all i.

01:08:22.490 --> 01:08:23.840
OK, does it make sense?

01:08:23.840 --> 01:08:28.984
Here what I am assuming is that
my population is homogeneous.

01:08:28.984 --> 01:08:29.609
Maybe it's not.

01:08:29.609 --> 01:08:31.484
Maybe I could actually
look at a finer grain,

01:08:31.484 --> 01:08:35.529
but I'm basically making a
statement about a population.

01:08:35.529 --> 01:08:41.670
And so maybe you kiss to the
left, and then you're not--

01:08:41.670 --> 01:08:44.010
I'm not making a statement
about a person individually,

01:08:44.010 --> 01:08:47.702
I'm making a statement about
the overall population.

01:08:47.702 --> 01:08:49.660
Now independence is
probably reasonable, right?

01:08:49.660 --> 01:08:53.080
This person just went
and know can seriously

01:08:53.080 --> 01:08:56.109
hope that these couples did not
communicate with each other.

01:08:56.109 --> 01:08:59.830
Or that you know Tanya did
not text that we should all

01:08:59.830 --> 01:09:01.810
turn our head to the left now.

01:09:01.810 --> 01:09:05.050
And there's no external
stimulus that forces people

01:09:05.050 --> 01:09:08.069
to do something different.

01:09:08.069 --> 01:09:15.810
OK, so-- sorry about that.

01:09:15.810 --> 01:09:19.899
Since we have about
less than 10 minutes.

01:09:19.899 --> 01:09:22.260
Let's do a little bit of
exercises, is that OK with you?

01:09:22.260 --> 01:09:24.950
So I just have some
exercises so we can see what

01:09:24.950 --> 01:09:26.580
an exercise going to look like.

01:09:26.580 --> 01:09:30.090
This is sort of similar to the
exercises you will see with me.

01:09:30.090 --> 01:09:31.674
We should do it together, OK?

01:09:31.674 --> 01:09:32.840
So now we're going to have--

01:09:32.840 --> 01:09:33.479
I have a test.

01:09:36.630 --> 01:09:42.950
So that's an exam
in probability.

01:09:42.950 --> 01:09:44.290
OK.

01:09:44.290 --> 01:09:50.899
And I'm going to have 15
students in this test.

01:09:50.899 --> 01:09:53.550
And hopefully, this
should be 15 grades

01:09:53.550 --> 01:09:57.210
that are representative of the
grades of all a large class.

01:09:57.210 --> 01:10:00.660
Right, so if you go you know
18.600, it's a large class,

01:10:00.660 --> 01:10:02.430
there's definitely
more than 15 students.

01:10:02.430 --> 01:10:04.750
And maybe, just by sampling
15 students at random,

01:10:04.750 --> 01:10:08.010
I want to have an idea of what
my grade distribution will

01:10:08.010 --> 01:10:09.090
look like.

01:10:09.090 --> 01:10:13.230
I'm grading them, I want
to make an educated guess.

01:10:13.230 --> 01:10:15.480
So I'm going to make
some modeling assumptions

01:10:15.480 --> 01:10:16.680
for those guys.

01:10:16.680 --> 01:10:22.830
So here, 15 students and
the grades are x1 to x15.

01:10:22.830 --> 01:10:26.050
Just like we had R1,
R2, all the way to R124.

01:10:26.050 --> 01:10:27.790
Those were my Ri's.

01:10:27.790 --> 01:10:29.610
And so now I have my xi's.

01:10:29.610 --> 01:10:33.270
And I'm going to
assume that xi follows

01:10:33.270 --> 01:10:39.060
a Gaussian or normal
distribution with min mu

01:10:39.060 --> 01:10:40.720
and variance sigma squared.

01:10:40.720 --> 01:10:43.710
Now this is modeling, right?

01:10:43.710 --> 01:10:45.720
Nobody told me there's
no physical process that

01:10:45.720 --> 01:10:46.471
makes this happen.

01:10:46.471 --> 01:10:48.761
We know that there's something
called the central limit

01:10:48.761 --> 01:10:50.720
theorem in the background
that says that things

01:10:50.720 --> 01:10:53.385
tend to be Gaussian, but this is
really a matter of convenience.

01:10:53.385 --> 01:10:55.090
Actually this is, if
you think about it,

01:10:55.090 --> 01:10:57.381
this is terrible because this
puts non-zero probability

01:10:57.381 --> 01:10:58.260
on negative scores.

01:10:58.260 --> 01:11:00.711
I'm definitely not going
to get a negative score.

01:11:00.711 --> 01:11:02.460
But you know it's good
enough because they

01:11:02.460 --> 01:11:05.310
know the probabilities
non-zero but it's probably 10

01:11:05.310 --> 01:11:06.240
to the minus 12.

01:11:06.240 --> 01:11:10.480
So I would be very unlucky
to see a negative score.

01:11:10.480 --> 01:11:24.740
So here's the list of grades, so
I have 65, 41, 70, 90, 58, 82,

01:11:24.740 --> 01:11:28.860
76, 78--

01:11:28.860 --> 01:11:35.660
maybe I should have done
it with 8 --59, 59--

01:11:35.660 --> 01:11:47.360
sitting next to each other
--84, 89, 134, 51, and 72.

01:11:47.360 --> 01:11:51.440
So those are the
scores that I got.

01:11:51.440 --> 01:11:53.930
There were clearly some
bonus points over there.

01:11:53.930 --> 01:12:05.134
And the question is,
find estimator for mu.

01:12:05.134 --> 01:12:06.300
What is my estimator for mu?

01:12:09.120 --> 01:12:11.070
Well, an estimator,
again, is something that

01:12:11.070 --> 01:12:12.510
depends on the random variable.

01:12:12.510 --> 01:12:15.180
All right, so mu is
the expectation, right?

01:12:15.180 --> 01:12:22.510
So a good estimator is
definitely the average score,

01:12:22.510 --> 01:12:24.460
just like we had the
average of the Ri's.

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

01:12:28.210 --> 01:12:31.060
going to boil down to being
a number of ones divided

01:12:31.060 --> 01:12:32.880
by the total numbers.

01:12:32.880 --> 01:12:41.070
Now if I'm looking
for an estimate,

01:12:41.070 --> 01:12:43.680
well, I need to actually
sum those numbers

01:12:43.680 --> 01:12:45.630
and divide them by 15.

01:12:45.630 --> 01:12:47.472
So my estimate is
going to be 1/15.

01:12:47.472 --> 01:12:49.430
Then I'm going to start
summing those numbers--

01:12:49.430 --> 01:12:51.430
65 plus 72.

01:12:54.320 --> 01:13:06.340
OK, and I can do
it, and it's 67.5.

01:13:06.340 --> 01:13:08.810
This is my estimate.

01:13:08.810 --> 01:13:13.810
Now if I want to compute
a standard deviation--

01:13:13.810 --> 01:13:18.690
so let's say estimate for sigma.

01:13:21.889 --> 01:13:23.180
You've seen that before, right?

01:13:23.180 --> 01:13:24.690
An estimate for sigma is what?

01:13:24.690 --> 01:13:27.150
An estimate for sigma, we'll
see methods to do this,

01:13:27.150 --> 01:13:31.360
but sigma squared
is the variance,

01:13:31.360 --> 01:13:35.941
or is the expectation, of x
minus expectation of x squared.

01:13:38.980 --> 01:13:40.480
And the problem is
that I don't know

01:13:40.480 --> 01:13:42.130
what those expectations are.

01:13:42.130 --> 01:13:47.260
And so I'm going to do what
99.9% percent of statistics is.

01:13:47.260 --> 01:13:49.710
And what is statistics about?

01:13:49.710 --> 01:13:51.350
What's my motto?

01:13:51.350 --> 01:13:54.590
Statistics is about replacing
expectations with averages.

01:13:54.590 --> 01:13:57.020
That's what all of
statistics is about.

01:13:57.020 --> 01:14:00.740
There's 300 pages in a purple
book called All of Statistics

01:14:00.740 --> 01:14:01.940
that tells you this.

01:14:01.940 --> 01:14:03.740
All right, and then
you do something fancy.

01:14:03.740 --> 01:14:05.570
Maybe you minimize
something after you

01:14:05.570 --> 01:14:07.100
replace the expectation.

01:14:07.100 --> 01:14:08.780
Maybe you need to
plug in other stuff.

01:14:08.780 --> 01:14:10.696
But really, every time
you see an expectation,

01:14:10.696 --> 01:14:12.110
you replace it by an average.

01:14:12.110 --> 01:14:13.470
OK let's do this.

01:14:13.470 --> 01:14:16.550
So sigma squared
hat will be what?

01:14:16.550 --> 01:14:20.420
It's going to be 1 over n,
sum from i equals 1 to n

01:14:20.420 --> 01:14:22.790
of xi minus--

01:14:22.790 --> 01:14:25.580
well, here I need to replace
my expectation by an average,

01:14:25.580 --> 01:14:27.230
which is really this average.

01:14:27.230 --> 01:14:31.600
I'm going to call
it mu hat squared.

01:14:31.600 --> 01:14:34.310
There, you have replaced my
expectation with average.

01:14:34.310 --> 01:14:38.150
OK so the golden thing
is, take your expectation

01:14:38.150 --> 01:14:39.460
and replace it with this.

01:14:45.580 --> 01:14:49.240
Frame it, get a tattoo, I don't
care but that's what it is.

01:14:49.240 --> 01:14:53.600
If you remember one thing from
this class, that's what it is.

01:14:53.600 --> 01:14:56.690
Now you can be fancy, if
you look at your calculator,

01:14:56.690 --> 01:14:59.660
it's going to put an n
minus 1 here because it

01:14:59.660 --> 01:15:00.535
wants to be unbiased.

01:15:00.535 --> 01:15:02.410
And those are things we
are going to come to.

01:15:02.410 --> 01:15:04.400
But let's say right
now we stick to this.

01:15:04.400 --> 01:15:06.440
And then when I
plug in my numbers.

01:15:06.440 --> 01:15:14.040
I'm going to get an
estimate for sigma,

01:15:14.040 --> 01:15:17.210
which is the square
root of the estimator

01:15:17.210 --> 01:15:18.810
once I plug in the numbers.

01:15:18.810 --> 01:15:21.622
And you can check that the
number, you get will be 18.

01:15:27.410 --> 01:15:31.060
So those are basic things and
if you've taken any AP stats

01:15:31.060 --> 01:15:32.810
this should be completely
standard to you.

01:15:35.800 --> 01:15:39.350
Now I have another list, and
I don't have time to see it.

01:15:42.029 --> 01:15:43.070
It doesn't really matter.

01:15:49.264 --> 01:15:50.430
OK, we'll do that next time.

01:15:50.430 --> 01:15:51.200
This is fine.

01:15:51.200 --> 01:15:55.241
We'll see another list
of numbers and see--

01:15:55.241 --> 01:15:57.240
we're going to think about
modeling assumptions.

01:15:57.240 --> 01:15:59.610
The goal of this exercise is
not to compute those things,

01:15:59.610 --> 01:16:01.610
it's really to think about
modeling assumptions.

01:16:01.610 --> 01:16:04.002
Is it reasonable to think
that things are IID?

01:16:04.002 --> 01:16:05.460
Is it reasonable
to think that they

01:16:05.460 --> 01:16:07.751
have all the same parameters,
that they're independent,

01:16:07.751 --> 01:16:08.880
et cetera,

01:16:08.880 --> 01:16:16.850
OK so one thing that I wanted
to add is, probably by tonight,

01:16:16.850 --> 01:16:18.810
so I will try to use--

01:16:18.810 --> 01:16:20.625
in the spirit of--

01:16:20.625 --> 01:16:22.250
I don't know what's
starting to happen.

01:16:22.250 --> 01:16:26.190
In the spirit of using
my iPad and fancy things,

01:16:26.190 --> 01:16:29.250
I will try to post some
videos of-- for in particular,

01:16:29.250 --> 01:16:33.840
who has never used a
statistical table to read, say,

01:16:33.840 --> 01:16:37.860
the quantiles of a
Gaussian distribution?

01:16:37.860 --> 01:16:39.480
OK, so there's several of you.

01:16:39.480 --> 01:16:42.600
This is a simple
but boring exercise.

01:16:42.600 --> 01:16:44.850
I will just post a
video on how to do this,

01:16:44.850 --> 01:16:46.785
and you will be able
to find it on Stellar.

01:16:46.785 --> 01:16:48.660
It's going to take five
minutes, and then you

01:16:48.660 --> 01:16:50.770
will know everything there
is to know about those things

01:16:50.770 --> 01:16:53.154
but that's something you need
for the first problem set.

01:16:53.154 --> 01:16:54.570
By the way, so the
problem set has

01:16:54.570 --> 01:16:57.180
30 exercises in probability.

01:16:57.180 --> 01:16:59.510
You need to do 15.

01:16:59.510 --> 01:17:01.470
And you only need to turn in 15.

01:17:01.470 --> 01:17:03.760
You can turn in all
of 30 if you want.

01:17:03.760 --> 01:17:07.230
But you need to know, by the
time we hit those things,

01:17:07.230 --> 01:17:08.760
you need to know--

01:17:08.760 --> 01:17:11.370
well actually, by next week you
need to know what's in there.

01:17:11.370 --> 01:17:13.970
So if you don't have time
to do all the homework,

01:17:13.970 --> 01:17:15.720
and then go back to
your probability class

01:17:15.720 --> 01:17:19.250
to figure out how to do it,
just do 15 easy that you can do.

01:17:19.250 --> 01:17:20.250
And return those things.

01:17:20.250 --> 01:17:21.827
But go back to your
probability class

01:17:21.827 --> 01:17:23.910
and make sure that you
know how to do all of them.

01:17:23.910 --> 01:17:25.650
Those are pretty
basic questions,

01:17:25.650 --> 01:17:28.290
and those are things that I'm
not going to slow down on.

01:17:28.290 --> 01:17:30.900
So you need to remember that
the expectation of the product

01:17:30.900 --> 01:17:32.316
of independent
random variables is

01:17:32.316 --> 01:17:34.560
a product of the expectations.

01:17:34.560 --> 01:17:36.810
Expectation of the sum, is
the sum of the expectation.

01:17:36.810 --> 01:17:38.775
This kind of thing,
which is a little silly,

01:17:38.775 --> 01:17:40.690
but it just requires
you practice.

01:17:40.690 --> 01:17:42.150
So, just have fun.

01:17:42.150 --> 01:17:43.440
Those are simple exercises.

01:17:43.440 --> 01:17:46.620
You will have fun remembering
your probability class.

01:17:46.620 --> 01:17:49.620
All right, so I'll
see you on Tuesday--

01:17:49.620 --> 01:17:51.410
or Monday.