jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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rLlZpnT02ZU	I'm actually stuck.
rLlZpnT02ZU	This A star is the one that actually maximizes
rLlZpnT02ZU	the integral of this function.
rLlZpnT02ZU	So we used the fact that for any function,
rLlZpnT02ZU	say delta, the integral over A of delta
rLlZpnT02ZU	is less than the integral over the set of X's
rLlZpnT02ZU	such that delta of X is non-negative of delta of X, dx.
rLlZpnT02ZU	And that's an obvious fact, just by picture, say.
rLlZpnT02ZU	And that's true for all A. Yeah?
rLlZpnT02ZU	AUDIENCE: [INAUDIBLE] could you use
rLlZpnT02ZU	like a portion under the axis as like less than
rLlZpnT02ZU	or equal to the portion above the axis?
rLlZpnT02ZU	PHILIPPE RIGOLLET: It's actually equal.
rLlZpnT02ZU	We know that the integral of f minus g--
rLlZpnT02ZU	the integral of delta is 0.
rLlZpnT02ZU	So there's actually exactly the same area above and below.
rLlZpnT02ZU	But yeah, you're right.
rLlZpnT02ZU	You could go to the extreme cases.
rLlZpnT02ZU	You're right.
rLlZpnT02ZU	No.
rLlZpnT02ZU	It's actually still be true, even if there was--
rLlZpnT02ZU	if this was a constant, that would still be true.
rLlZpnT02ZU	Here, I never use the fact that the integral is equal to 0.
rLlZpnT02ZU	I could shift this function by 1 so that the integral of delta
rLlZpnT02ZU	is equal to 1, and it would still
rLlZpnT02ZU	be true that it's maximized when I take A to be
rLlZpnT02ZU	the set where it's positive.
rLlZpnT02ZU	Just need to make sure that there is someplace where it is,
rLlZpnT02ZU	but that's about it.
rLlZpnT02ZU	Of course, we used this before, when we made this thing.
rLlZpnT02ZU	But just the last argument, this last fact
rLlZpnT02ZU	does not require that.
rLlZpnT02ZU	All right.
rLlZpnT02ZU	So now we have this notion of--
rLlZpnT02ZU	I need the--
rLlZpnT02ZU	OK.
rLlZpnT02ZU	So we have this notion of distance
rLlZpnT02ZU	between probability measures.
rLlZpnT02ZU	I mean, these things are exactly what--
rLlZpnT02ZU	if I were to be in a formal math class and I said,
rLlZpnT02ZU	here are the axioms that a distance should satisfy,
rLlZpnT02ZU	those are exactly those things.
rLlZpnT02ZU	If it's not satisfying this thing,
rLlZpnT02ZU	it's called pseudo-distance or quasi-distance or just metric
rLlZpnT02ZU	or nothing at all, honestly.
rLlZpnT02ZU	So it's a distance.
rLlZpnT02ZU	It's symmetric, non-negative, equal to 0,
rLlZpnT02ZU	if and only if the two arguments are equal, then
rLlZpnT02ZU	it satisfies the triangle inequality.
rLlZpnT02ZU	And so that means that we have this actual total variation
rLlZpnT02ZU	distance between probability distributions.
rLlZpnT02ZU	And here is now a statistical strategy to implement our goal.
rLlZpnT02ZU	Remember, our goal was to spit out
rLlZpnT02ZU	a theta hat, which was close such that P theta
rLlZpnT02ZU	hat was close to P theta star.
rLlZpnT02ZU	So hopefully, we were trying to minimize the total variation
rLlZpnT02ZU	distance between P theta hat and P theta star.
rLlZpnT02ZU	Now, we cannot do that, because just by this fact, this slide,
rLlZpnT02ZU	if we wanted to do that directly, we would just take--
rLlZpnT02ZU	well, let's take theta hat equals theta star and that will
rLlZpnT02ZU	give me the value 0.
rLlZpnT02ZU	And that's the minimum possible value we can take.
rLlZpnT02ZU	The problem is that we don't know
rLlZpnT02ZU	what the total variation is to something that we don't know.
rLlZpnT02ZU	We know how to compute total variations if I give you
rLlZpnT02ZU	the two arguments.
rLlZpnT02ZU	But here, one of the arguments is not known.
rLlZpnT02ZU	P theta star is not known to us, so we need to estimate it.
rLlZpnT02ZU	And so here is the strategy.
rLlZpnT02ZU	Just build an estimator of the total variation
rLlZpnT02ZU	distance between P theta and P theta star
rLlZpnT02ZU	for all candidate theta, all possible theta
rLlZpnT02ZU	in capital theta.
rLlZpnT02ZU	Now, if this is a good estimate, then when I minimize it,
rLlZpnT02ZU	I should get something that's close to P theta star.
rLlZpnT02ZU	So here's the strategy.
rLlZpnT02ZU	This is my function that maps theta
rLlZpnT02ZU	to the total variation between P theta and P theta star.
rLlZpnT02ZU	I know it's minimized at theta star.
rLlZpnT02ZU	That's definitely TV of P-- and the value here, the y-axis
rLlZpnT02ZU	should say 0.
rLlZpnT02ZU	And so I don't know this guy, so I'm
rLlZpnT02ZU	going to estimate it by some estimator that
rLlZpnT02ZU	comes from my data.
rLlZpnT02ZU	Hopefully, the more data I have, the better this estimator is.
rLlZpnT02ZU	And I'm going to try to minimize this estimator now.
rLlZpnT02ZU	And if the two things are close, then the minima
rLlZpnT02ZU	should be close.
rLlZpnT02ZU	That's a pretty good estimation strategy.
rLlZpnT02ZU	The problem is that it's very unclear
rLlZpnT02ZU	how you would build this estimator of TV,
rLlZpnT02ZU	of the Total Variation.
rLlZpnT02ZU	So building estimators, as I said,
rLlZpnT02ZU	typically consists in replacing expectations by averages.
rLlZpnT02ZU	But there's no simple way of expressing the total variation
rLlZpnT02ZU	distance as the expectations with respect
rLlZpnT02ZU	to theta star of anything.
rLlZpnT02ZU	So what we're going to do is we're
rLlZpnT02ZU	going to move from total variation distance
rLlZpnT02ZU	to another notion of distance that sort of has

rLlZpnT02ZU

I'm actually stuck.

rLlZpnT02ZU

This A star is the one that actually maximizes

rLlZpnT02ZU

the integral of this function.

rLlZpnT02ZU

So we used the fact that for any function,

rLlZpnT02ZU

say delta, the integral over A of delta

rLlZpnT02ZU

is less than the integral over the set of X's

rLlZpnT02ZU

such that delta of X is non-negative of delta of X, dx.

rLlZpnT02ZU

And that's an obvious fact, just by picture, say.

rLlZpnT02ZU

And that's true for all A. Yeah?

rLlZpnT02ZU

AUDIENCE: [INAUDIBLE] could you use

rLlZpnT02ZU

like a portion under the axis as like less than

rLlZpnT02ZU

or equal to the portion above the axis?

rLlZpnT02ZU

PHILIPPE RIGOLLET: It's actually equal.

rLlZpnT02ZU

We know that the integral of f minus g--

rLlZpnT02ZU

the integral of delta is 0.

rLlZpnT02ZU

So there's actually exactly the same area above and below.

rLlZpnT02ZU

But yeah, you're right.

rLlZpnT02ZU

You could go to the extreme cases.

rLlZpnT02ZU

You're right.

rLlZpnT02ZU

No.

rLlZpnT02ZU

It's actually still be true, even if there was--

rLlZpnT02ZU

if this was a constant, that would still be true.

rLlZpnT02ZU

Here, I never use the fact that the integral is equal to 0.

rLlZpnT02ZU

I could shift this function by 1 so that the integral of delta

rLlZpnT02ZU

is equal to 1, and it would still

rLlZpnT02ZU

be true that it's maximized when I take A to be

rLlZpnT02ZU

the set where it's positive.

rLlZpnT02ZU

Just need to make sure that there is someplace where it is,

rLlZpnT02ZU

but that's about it.

rLlZpnT02ZU

Of course, we used this before, when we made this thing.

rLlZpnT02ZU

But just the last argument, this last fact

rLlZpnT02ZU

does not require that.

rLlZpnT02ZU

All right.

rLlZpnT02ZU

So now we have this notion of--

rLlZpnT02ZU

I need the--

rLlZpnT02ZU

OK.

rLlZpnT02ZU

So we have this notion of distance

rLlZpnT02ZU

between probability measures.

rLlZpnT02ZU

I mean, these things are exactly what--

rLlZpnT02ZU

if I were to be in a formal math class and I said,

rLlZpnT02ZU

here are the axioms that a distance should satisfy,

rLlZpnT02ZU

those are exactly those things.

rLlZpnT02ZU

If it's not satisfying this thing,

rLlZpnT02ZU

it's called pseudo-distance or quasi-distance or just metric

rLlZpnT02ZU

or nothing at all, honestly.

rLlZpnT02ZU

So it's a distance.

rLlZpnT02ZU

It's symmetric, non-negative, equal to 0,

rLlZpnT02ZU

if and only if the two arguments are equal, then

rLlZpnT02ZU

it satisfies the triangle inequality.

rLlZpnT02ZU

And so that means that we have this actual total variation

rLlZpnT02ZU

distance between probability distributions.

rLlZpnT02ZU

And here is now a statistical strategy to implement our goal.

rLlZpnT02ZU

Remember, our goal was to spit out

rLlZpnT02ZU

a theta hat, which was close such that P theta

rLlZpnT02ZU

hat was close to P theta star.

rLlZpnT02ZU

So hopefully, we were trying to minimize the total variation

rLlZpnT02ZU

distance between P theta hat and P theta star.

rLlZpnT02ZU

Now, we cannot do that, because just by this fact, this slide,

rLlZpnT02ZU

if we wanted to do that directly, we would just take--

rLlZpnT02ZU

well, let's take theta hat equals theta star and that will

rLlZpnT02ZU

give me the value 0.

rLlZpnT02ZU

And that's the minimum possible value we can take.

rLlZpnT02ZU

The problem is that we don't know

rLlZpnT02ZU

what the total variation is to something that we don't know.

rLlZpnT02ZU

We know how to compute total variations if I give you

rLlZpnT02ZU

the two arguments.

rLlZpnT02ZU

But here, one of the arguments is not known.

rLlZpnT02ZU

P theta star is not known to us, so we need to estimate it.

rLlZpnT02ZU

And so here is the strategy.

rLlZpnT02ZU

Just build an estimator of the total variation

rLlZpnT02ZU

distance between P theta and P theta star

rLlZpnT02ZU

for all candidate theta, all possible theta

rLlZpnT02ZU

in capital theta.

rLlZpnT02ZU

Now, if this is a good estimate, then when I minimize it,

rLlZpnT02ZU

I should get something that's close to P theta star.

rLlZpnT02ZU

So here's the strategy.

rLlZpnT02ZU

This is my function that maps theta

rLlZpnT02ZU

to the total variation between P theta and P theta star.

rLlZpnT02ZU

I know it's minimized at theta star.

rLlZpnT02ZU

That's definitely TV of P-- and the value here, the y-axis

rLlZpnT02ZU

should say 0.

rLlZpnT02ZU

And so I don't know this guy, so I'm

rLlZpnT02ZU

going to estimate it by some estimator that

rLlZpnT02ZU

comes from my data.

rLlZpnT02ZU

Hopefully, the more data I have, the better this estimator is.

rLlZpnT02ZU

And I'm going to try to minimize this estimator now.

rLlZpnT02ZU

And if the two things are close, then the minima

rLlZpnT02ZU

should be close.

rLlZpnT02ZU

That's a pretty good estimation strategy.

rLlZpnT02ZU

The problem is that it's very unclear

rLlZpnT02ZU

how you would build this estimator of TV,

rLlZpnT02ZU

of the Total Variation.

rLlZpnT02ZU

So building estimators, as I said,

rLlZpnT02ZU

typically consists in replacing expectations by averages.

rLlZpnT02ZU

But there's no simple way of expressing the total variation

rLlZpnT02ZU

distance as the expectations with respect

rLlZpnT02ZU

to theta star of anything.

rLlZpnT02ZU

So what we're going to do is we're

rLlZpnT02ZU

going to move from total variation distance

rLlZpnT02ZU

to another notion of distance that sort of has