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rLlZpnT02ZU | And if you've
heard of KL, you've |
rLlZpnT02ZU | probably heard of entropy. |
rLlZpnT02ZU | And that's what-- it's
basically minus the entropy. |
rLlZpnT02ZU | And that's a quantity that
just depends on theta star. |
rLlZpnT02ZU | But it's just the number. |
rLlZpnT02ZU | I could compute this
number if I told |
rLlZpnT02ZU | you this is n theta star 1. |
rLlZpnT02ZU | You could compute this. |
rLlZpnT02ZU | So now I'm going
to try to minimize |
rLlZpnT02ZU | the estimate of this function. |
rLlZpnT02ZU | And minimizing a function or
a function plus a constant |
rLlZpnT02ZU | is the same thing. |
rLlZpnT02ZU | I'm just shifting the
function here or here, |
rLlZpnT02ZU | but it's the same minimizer. |
rLlZpnT02ZU | OK. |
rLlZpnT02ZU | So the function that maps
theta to KL of P theta star |
rLlZpnT02ZU | to P theta is of the form
constant minus this expectation |
rLlZpnT02ZU | of a log of P theta. |
rLlZpnT02ZU | Everybody agrees? |
rLlZpnT02ZU | Are there any
questions about this? |
rLlZpnT02ZU | Are there any
remarks, including I |
rLlZpnT02ZU | have no idea what's
happening right now? |
rLlZpnT02ZU | OK. |
rLlZpnT02ZU | We're good? |
rLlZpnT02ZU | Yeah. |
rLlZpnT02ZU | AUDIENCE: So when you're
actually employing this method, |
rLlZpnT02ZU | how do you know which theta
to use as theta star and which |
rLlZpnT02ZU | isn't? |
rLlZpnT02ZU | PHILIPPE RIGOLLET: So this is
not a method just yet, right? |
rLlZpnT02ZU | I'm just describing to
you what the KL divergence |
rLlZpnT02ZU | between two distributions is. |
rLlZpnT02ZU | If you really wanted
to compute it, |
rLlZpnT02ZU | you would need to know
what P theta star is |
rLlZpnT02ZU | and what P theta is. |
rLlZpnT02ZU | AUDIENCE: Right. |
rLlZpnT02ZU | PHILIPPE RIGOLLET: And so here,
I'm just saying at some point, |
rLlZpnT02ZU | we still-- so here, you see-- |
rLlZpnT02ZU | so now let's move onto one step. |
rLlZpnT02ZU | I don't know expectation
of theta star. |
rLlZpnT02ZU | But I have data that comes
from distribution P theta star. |
rLlZpnT02ZU | So the expectation by
the law of large numbers |
rLlZpnT02ZU | should be close to the average. |
rLlZpnT02ZU | And so what I'm doing
is I'm replacing any-- |
rLlZpnT02ZU | I can actually-- this is a very
standard estimation method. |
rLlZpnT02ZU | You write something as an
expectation with respect |
rLlZpnT02ZU | to the data-generating
process of some function. |
rLlZpnT02ZU | And then you replace this by
the average of this function. |
rLlZpnT02ZU | And the law of large
numbers tells me |
rLlZpnT02ZU | that those two quantities
should actually be close. |
rLlZpnT02ZU | Now, it doesn't mean that's
going to be the end of the day, |
rLlZpnT02ZU | right. |
rLlZpnT02ZU | When we did Xn bar, that
was the end of the day. |
rLlZpnT02ZU | We had an expectation. |
rLlZpnT02ZU | We replaced it by an average. |
rLlZpnT02ZU | And then we were gone. |
rLlZpnT02ZU | But here, we still
have to do something, |
rLlZpnT02ZU | because this is not
telling me what theta is. |
rLlZpnT02ZU | Now I still have to
minimize this average. |
rLlZpnT02ZU | So this is now my candidate
estimator for KL, KL hat. |
rLlZpnT02ZU | And that's the one
where I said, well, it's |
rLlZpnT02ZU | going to be of the
form of constant. |
rLlZpnT02ZU | And this constant, I don't know. |
rLlZpnT02ZU | You're right. |
rLlZpnT02ZU | I have no idea what
this constant is. |
rLlZpnT02ZU | It depends on P theta star. |
rLlZpnT02ZU | But then I have minus something
that I can completely compute. |
rLlZpnT02ZU | If you give me data and theta,
I can compute this entire thing. |
rLlZpnT02ZU | And now what I claim is that
the minimizer of f or f plus-- |
rLlZpnT02ZU | f of X or f of X plus
4 are the same thing, |
rLlZpnT02ZU | or say 4 plus f of
X. I'm just shifting |
rLlZpnT02ZU | the plot of my
function up and down, |
rLlZpnT02ZU | but the minimizer stays
exactly where it is. |
rLlZpnT02ZU | If I have a function-- |
rLlZpnT02ZU | so now I have a
function of theta. |
rLlZpnT02ZU | This is KL hat of P
theta star, P theta. |
rLlZpnT02ZU | And it's of the form--
it's a function like this. |
rLlZpnT02ZU | I don't know where
this function is. |
rLlZpnT02ZU | It might very well be this
function or this function. |
rLlZpnT02ZU | Every time it's a translation
on the y-axis of all these guys. |
rLlZpnT02ZU | And the value that I translated
by depends on theta star. |
rLlZpnT02ZU | I don't know what it is. |
rLlZpnT02ZU | But what I claim is that the
minimizer is always this guy, |
rLlZpnT02ZU | regardless of what the value is. |
rLlZpnT02ZU | OK? |
rLlZpnT02ZU | So when I say constant, it's a
constant with respect to theta. |
rLlZpnT02ZU | It's an unknown constant. |
rLlZpnT02ZU | But it's with respect to theta,
so without loss of generality, |
rLlZpnT02ZU | I can assume that this
constant is 0 for my purposes, |
rLlZpnT02ZU | or 25 if you prefer. |
rLlZpnT02ZU | All right. |
rLlZpnT02ZU | So we'll just keep going
on this property next time. |
rLlZpnT02ZU | And we'll see how from
here we can move on to-- |
rLlZpnT02ZU | the likelihood is actually going
to come out of this formula. |
rLlZpnT02ZU | Thanks. |
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