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RICHARD DE NEUFVILLE: So now,
what if there are many stages?

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So you just had a
2-stage problem.

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And the general idea
here is, all right--

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I mean, a 2-step and one
choice that is outcomes.

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If I want to do more, so for
example, in the same question,

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suppose that you are in--

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as a student, you were wondering
whether to take the raincoat.

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Suppose you decide to
consult The Weather Channel.

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So again, for simplicity, let's
suppose it has a yes/no answer,

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it's going to rain or not.

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In practice, of
course, they give you

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a percentage of, it will
have a 40% chance of rain

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or something, which means
you can never tell if they're

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right or wrong unless you study
their results over a long,

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long period.

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So it's a yes/no answer.

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And of course, now,
these forecasts

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say it rains are wrong.

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That is, they can be wrong.

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But if they do say
rain, you're saying,

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hmm, I thought it was
a 30% chance of rain,

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but now, hmm, they say rain,
so I'm going to bump it up.

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Just because they said it will
rain doesn't mean it will rain,

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but it's likely to change it.

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So you're going to have
to have new probabilities.

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After you have the
information, you

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revise your prior probabilities.

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So let me show you
what this looks like.

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So I've now set it up on a
spreadsheet kind of basis here.

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And I either have I don't--

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I don't look at The Weather
Channel and I have no news,

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so that was what I
had before, my best

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choice at a value of 0.8.

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But if I look at
The Weather Channel,

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it will be, say, a forecast rain
or not, in this simple case.

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Now, you have to say,
what is the probability

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that the forecast is correct?

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Well, to do that,
you're going to have

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to have some kind of
estimate of the performance

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of that information.

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So you don't really
know what the chances

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are they will say rain or not
rain, so that's an unknown.

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But then, you come back
to where you were before,

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but now, this is the setup
for the basic problem

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before you had The
Weather Channel.

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You now have, all right, yeah,
I still have rain or no rain,

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but this time, the
probabilities of rain or no rain

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depend upon the forecast.

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So if I looked at it
was a rain forecast,

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the probability of rain or
no rain has been changed.

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It's more likely to rain.

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You believe it's
more likely to rain.

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So this is now
different than if it

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said it's going to be a
clear day, in which case,

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you'd have different ones.

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So the problem, given that
you have a second stage here,

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some further
information gathering,

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you're going to
have to calculate

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all these probabilities.

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And therefore, the new
expected values of difference,

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and then if the outcomes
here are different,

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and the probabilities are
different, so that you have--

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adding on a section will
change a lot, basically.

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It changes all your information.

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Technically, it's
called that you

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had the prior probabilities,
what you started out

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with, the 0.3 and 0.7
here, and then, you

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had what's known as the
posterior probabilities.

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And this doesn't
refer to your tush.

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It's just the after
the fact part.

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After the fact, the
fact being that you

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had some kind of information, in
this case, The Weather Channel

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report, which revised your
prior probabilities and made you

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go through.

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So now, what you
have to have is that

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changes your
decisions, of course.

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That's expected.

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And the calculations
get complicated.

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In general, you have
to go through something

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called a Bayesian update, which
I'm not going to explain here,

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because that's not
the point of this

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and we're not going to
go into that detail.

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But this is a
complicated process.

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That's the point I
wanted to mention.

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The second aspect
is, given that you

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have all these probabilities,
how do you solve the problem?

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Well, I'll just
outline it for you.

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And it's an example
of what is known

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as dynamic programming, which
is not part of the course.

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But for those of you who
may have heard about it,

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it gives you a hook to go on.

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If not, let me just--

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you haven't heard
of it, that's fine.

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But it basically says that
the approach is to say,

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if I were at the last stage, and
I knew what these values were,

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I could find out what
would be the better

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one under this
circumstances, and I

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would choose the better
of these and use that

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to calculate the outcome here.

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In other words, you'd work
your way back through the tree,

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starting at the end, and
step by step, phase by phase,

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step by step, you'd--

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OK, if I were at the
end, I would choose, say,

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the raincoat.

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The other case, I might not.

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And derive the
outcome, and now, I

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would be able to calculate it.

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So it's the term of art
used, it's folding back.

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It starts at the end
and works your way

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through the analysis
to find out,

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what is your better choice?

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Why am I telling you this?

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I'm telling you this
because what the message is,

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that in order to do the
decision analysis for anything

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but the simplest case,
it isn't a simple thing.

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You have to go through
a lot of calculations.

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And in order to do
it, you have to have

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a lot of more information
on the relative accuracy

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of The Weather Channel
forecast, for example.

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And you have to go through
a Bayesian analysis

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to figure out the
answers to these unknowns

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here, and so forth.

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It isn't easy to set up.

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I can teach this
stuff, how to do it.

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I've done it before.

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But it's a painful process.

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And here, I've only shown
a most trivial situation,

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that is, only have two choices,
two possible outcomes, two

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possible responses from
The Weather Channel.

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It gets quite complicated.

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So just imagine that
instead of-- this is for two

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choices, two
outcomes, two stages.

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Just a match extension
of three of each,

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and you're now talking about
a really, really complicated.

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And I have, just to set up--
and I haven't deliberately.

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I haven't tried to
show you the way

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to calculate the revisions
or the prior probability

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to learn about the future.

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