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rLlZpnT02ZU
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The KL divergence
is non-negative.
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rLlZpnT02ZU
|
Who knows the Jensen's
inequality here?
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rLlZpnT02ZU
|
That should be a subset
of the people who
|
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|
raised their hand when I asked
what a convex function is.
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
So you know what
Jensen's inequality is.
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rLlZpnT02ZU
|
This is Jensen's-- the
proof is just one step
|
rLlZpnT02ZU
|
Jensen's inequality, which
we will not go into details.
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rLlZpnT02ZU
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But that's basically
an inequality
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|
involving expectation
of a convex function
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of a random variable compared
to the convex function
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|
of the expectation
of a random variable.
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rLlZpnT02ZU
|
If you know Jensen,
have fun and prove it.
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rLlZpnT02ZU
|
What's really nice is that
if the KL is equal to 0,
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then the two distributions
are the same.
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And that's something
we're looking for.
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Everything else we're
happy to throw out.
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And actually, if
you pay attention,
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we're actually really
throwing out everything else.
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rLlZpnT02ZU
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So they're not symmetric.
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rLlZpnT02ZU
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It does satisfy the triangle
inequality in general.
|
rLlZpnT02ZU
|
But it's non-negative and
it's 0 if and only if the two
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distributions are the same.
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rLlZpnT02ZU
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And that's all we care about.
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rLlZpnT02ZU
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And that's what we call
a divergence rather than
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a distance, and divergence will
be enough for our purposes.
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rLlZpnT02ZU
|
And actually, this
asymmetry, the fact
|
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|
that it's not flipping--
the first time I saw it,
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rLlZpnT02ZU
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I was just annoyed.
|
rLlZpnT02ZU
|
I was like, can we
just like, I don't
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rLlZpnT02ZU
|
know, take the average
of the KL between P theta
|
rLlZpnT02ZU
|
and P theta prime and P
theta prime and P theta,
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rLlZpnT02ZU
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you would think maybe
you could do this.
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rLlZpnT02ZU
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You just symmatrize it by just
taking the average of the two
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possible values it can take.
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rLlZpnT02ZU
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The problem is that this will
still not satisfy the triangle
|
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|
inequality.
|
rLlZpnT02ZU
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And there's no way basically
to turn it into something
|
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|
that is a distance.
|
rLlZpnT02ZU
|
But the divergence is doing
a pretty good thing for us.
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And this is what will allow us
to estimate it and basically
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|
overcome what we could not
do with the total variation.
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rLlZpnT02ZU
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So the first thing
that you want to notice
|
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|
is the total
variation distance--
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the KL divergence,
sorry, is actually
|
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an expectation of something.
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rLlZpnT02ZU
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Look at what it is here.
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rLlZpnT02ZU
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It's the integral of some
function against a density.
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rLlZpnT02ZU
|
That's exactly the definition
of an expectation, right?
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rLlZpnT02ZU
|
So this is the expectation
of this particular function
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|
with respect to this density f.
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So in particular, if I call
this is density f-- if I say,
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I want the true distribution
to be the first argument,
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this is an expectation
with respect
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to the true distribution from
which my data is actually
|
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|
drawn of the log of this ratio.
|
rLlZpnT02ZU
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So ha ha.
|
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I'm a statistician.
|
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Now I have an expectation.
|
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I can replace it by an
average, because I have data
|
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|
from this distribution.
|
rLlZpnT02ZU
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And I could actually replace
the expectation by an average
|
rLlZpnT02ZU
|
and try to minimize here.
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rLlZpnT02ZU
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The problem is that--
|
rLlZpnT02ZU
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actually the star here should
be in front of the theta,
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|
not of the P, right?
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rLlZpnT02ZU
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That's P theta star,
not P star theta.
|
rLlZpnT02ZU
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But here, I still
cannot compute it,
|
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because I have this P
theta star that shows up.
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I don't know what it is.
|
rLlZpnT02ZU
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And that's now where
the log plays a role.
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rLlZpnT02ZU
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If you actually pay
attention, I said
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you can use Jensen to
prove all this stuff.
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rLlZpnT02ZU
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You could actually replace the
log by any concave function.
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rLlZpnT02ZU
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That would be f divergent.
|
rLlZpnT02ZU
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That's called an f divergence.
|
rLlZpnT02ZU
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But the log itself is a
very, very specific property,
|
rLlZpnT02ZU
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which allows us to say
that the log of the ratio
|
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is the ratio of the log.
|
rLlZpnT02ZU
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Now, this thing here
does not depend on theta.
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rLlZpnT02ZU
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If I think of this KL divergence
as a function of theta,
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then the first part is
actually a constant.
|
rLlZpnT02ZU
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If I change theta, this thing
is never going to change.
|
rLlZpnT02ZU
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It depends only on theta star.
|
rLlZpnT02ZU
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So if I look at
this function KL--
|
rLlZpnT02ZU
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so if I look at the
function, theta maps
|
rLlZpnT02ZU
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to KL P theta
star, P theta, it's
|
rLlZpnT02ZU
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of the form expectation
with respect to theta star,
|
rLlZpnT02ZU
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log of P theta star
of X. And then I
|
rLlZpnT02ZU
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have minus expectation with
respect to theta star of log
|
rLlZpnT02ZU
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of P theta of x.
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rLlZpnT02ZU
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Now as I said, this thing
here, this second expectation
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rLlZpnT02ZU
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is a function of theta.
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When theta changes, this
thing is going to change.
|
rLlZpnT02ZU
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And that's a good thing.
|
rLlZpnT02ZU
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We want something that reflects
how close theta and theta
|
rLlZpnT02ZU
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star are.
|
rLlZpnT02ZU
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But this thing is
not going to change.
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rLlZpnT02ZU
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This is a fixed value.
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rLlZpnT02ZU
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Actually, it's the negative
entropy of P theta star.
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