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rLlZpnT02ZU | the same properties
and the same feeling |
rLlZpnT02ZU | and the same motivations as
the total variation distance. |
rLlZpnT02ZU | But for this guy, we
will be able to build |
rLlZpnT02ZU | an estimate for it,
because it's actually |
rLlZpnT02ZU | going to be of the form
expectation of something. |
rLlZpnT02ZU | And we're going to
be able to replace |
rLlZpnT02ZU | the expectation by an average
and then minimize this average. |
rLlZpnT02ZU | So this surrogate for
total variation distance |
rLlZpnT02ZU | is actually called the
Kullback-Leibler divergence. |
rLlZpnT02ZU | And why we call it divergence
is because it's actually |
rLlZpnT02ZU | not a distance. |
rLlZpnT02ZU | It's not going to be
symmetric to start with. |
rLlZpnT02ZU | So this Kullback-Leibler
or even KL divergence-- |
rLlZpnT02ZU | I will just refer to it as KL-- |
rLlZpnT02ZU | is actually just
more convenient. |
rLlZpnT02ZU | But it has some roots coming
from information theory, which |
rLlZpnT02ZU | I will not delve into. |
rLlZpnT02ZU | But if any of you is
actually a Core 6 student, |
rLlZpnT02ZU | I'm sure you've
seen that in some-- |
rLlZpnT02ZU | I don't know-- course that
has any content on information |
rLlZpnT02ZU | theory. |
rLlZpnT02ZU | All right. |
rLlZpnT02ZU | So the KL divergence between two
probability measures, P theta |
rLlZpnT02ZU | and P theta prime-- |
rLlZpnT02ZU | and here, as I said, it's not
going to be the symmetric, |
rLlZpnT02ZU | so it's very important
for you to specify |
rLlZpnT02ZU | which order you say it is,
between P theta and P theta |
rLlZpnT02ZU | prime. |
rLlZpnT02ZU | It's different from saying
between P theta prime and P |
rLlZpnT02ZU | theta. |
rLlZpnT02ZU | And so we denote it by KL. |
rLlZpnT02ZU | And so remember, before we had
either the sum or the integral |
rLlZpnT02ZU | of 1/2 of the distance--
absolute value of the distance |
rLlZpnT02ZU | between the PMFs and 1/2
of the absolute values |
rLlZpnT02ZU | of the distances between the
probability density functions. |
rLlZpnT02ZU | And then we replace
this absolute value |
rLlZpnT02ZU | of the distance divided by
2 by this weird function. |
rLlZpnT02ZU | This function is P
theta, log P theta, |
rLlZpnT02ZU | divided by P theta prime. |
rLlZpnT02ZU | That's the function. |
rLlZpnT02ZU | That's a weird function. |
rLlZpnT02ZU | OK. |
rLlZpnT02ZU | So this was what we had. |
rLlZpnT02ZU | That's the TV. |
rLlZpnT02ZU | And the KL, if I use the
same notation, f and g, |
rLlZpnT02ZU | is integral of f of X, log
of f of X over g of X, dx. |
rLlZpnT02ZU | It's a bit different. |
rLlZpnT02ZU | And I go from discrete to
continuous using an integral. |
rLlZpnT02ZU | Everybody can read this. |
rLlZpnT02ZU | Everybody's fine with this. |
rLlZpnT02ZU | Is there any uncertainty about
the actual definition here? |
rLlZpnT02ZU | So here I go straight
to the definition, |
rLlZpnT02ZU | which is just
plugging the functions |
rLlZpnT02ZU | into some integral and compute. |
rLlZpnT02ZU | So I don't bother with
maxima or anything. |
rLlZpnT02ZU | I mean, there is
something like that, |
rLlZpnT02ZU | but it's certainly not as
natural as the total variation. |
rLlZpnT02ZU | Yes? |
rLlZpnT02ZU | AUDIENCE: The total
variation, [INAUDIBLE].. |
rLlZpnT02ZU | PHILIPPE RIGOLLET:
Yes, just because it's |
rLlZpnT02ZU | hard to build anything
from total variation, |
rLlZpnT02ZU | because I don't know it. |
rLlZpnT02ZU | So it's very difficult.
But if you can actually-- |
rLlZpnT02ZU | and even computing it
between two Gaussians, |
rLlZpnT02ZU | just try it for yourself. |
rLlZpnT02ZU | And please stop doing it
after at most six minutes, |
rLlZpnT02ZU | because you won't
be able to do it. |
rLlZpnT02ZU | And so it's just very
hard to manipulate, |
rLlZpnT02ZU | like this integral of
absolute values of differences |
rLlZpnT02ZU | between probability
density function, at least |
rLlZpnT02ZU | for the probability
density functions |
rLlZpnT02ZU | we're used to manipulate
is actually a nightmare. |
rLlZpnT02ZU | And so people prefer KL,
because for the Gaussian, |
rLlZpnT02ZU | this is going to be theta
minus theta prime squared. |
rLlZpnT02ZU | And then we're
going to be happy. |
rLlZpnT02ZU | And so those things are
much easier to manipulate. |
rLlZpnT02ZU | But it's really--
the total variation |
rLlZpnT02ZU | is telling you how
far in the worst case |
rLlZpnT02ZU | the two probabilities can be. |
rLlZpnT02ZU | This is really the
intrinsic notion |
rLlZpnT02ZU | of closeness between
probabilities. |
rLlZpnT02ZU | So that's really the
one-- if we could, |
rLlZpnT02ZU | that's the one we
would go after. |
rLlZpnT02ZU | Sometimes people will
compute them numerically, |
rLlZpnT02ZU | so that they can say, oh, here's
the total variation distance I |
rLlZpnT02ZU | have between those two things. |
rLlZpnT02ZU | And then you actually
know that that |
rLlZpnT02ZU | means they are close, because
the absolute value-- if I tell |
rLlZpnT02ZU | you total variation is
0.01, like we did here, |
rLlZpnT02ZU | it has a very specific meaning. |
rLlZpnT02ZU | If I tell you the KL
divergence is 0.01, |
rLlZpnT02ZU | it's not clear what it means. |
rLlZpnT02ZU | OK. |
rLlZpnT02ZU | So what are the properties? |
rLlZpnT02ZU | The KL divergence between
P theta and P theta prime |
rLlZpnT02ZU | is different from the KL
divergence between P theta |
rLlZpnT02ZU | prime and P theta in general. |
rLlZpnT02ZU | Of course, in general,
because if theta |
rLlZpnT02ZU | is equal to theta prime,
then this certainly is true. |
rLlZpnT02ZU | So there's cases
when it's not true. |
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