jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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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.

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.