jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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rLlZpnT02ZU	PHILIPPE RIGOLLET: So that's not Slutsky, right?
rLlZpnT02ZU	AUDIENCE: That's [INAUDIBLE].
rLlZpnT02ZU	PHILIPPE RIGOLLET: So Slutsky tells you that if you--
rLlZpnT02ZU	Slutsky's about combining two types of convergence.
rLlZpnT02ZU	So Slutsky tells you that if you actually
rLlZpnT02ZU	have one Xn that converges to X in distribution and Yn
rLlZpnT02ZU	that converges to Y in probability, then
rLlZpnT02ZU	you can actually multiply Xn and Yn
rLlZpnT02ZU	and get that the limit in distribution
rLlZpnT02ZU	is the product of X and Y, where X is now a constant.
rLlZpnT02ZU	And here we have the constant, which is 1.
rLlZpnT02ZU	But I did that already, right?
rLlZpnT02ZU	Using Slutsky to replace it for the--
rLlZpnT02ZU	to replace P by Xn bar, we've done
rLlZpnT02ZU	that last time, maybe a couple of times ago, actually.
rLlZpnT02ZU	Yeah.
rLlZpnT02ZU	AUDIENCE: So I guess these statements are [INAUDIBLE]..
rLlZpnT02ZU	PHILIPPE RIGOLLET: That's correct.
rLlZpnT02ZU	AUDIENCE: So could we like figure out [INAUDIBLE]
rLlZpnT02ZU	can we set a finite [INAUDIBLE].
rLlZpnT02ZU	PHILIPPE RIGOLLET: So of course, the short answer is no.
rLlZpnT02ZU	So here's how you would go about thinking
rLlZpnT02ZU	about which method is better.
rLlZpnT02ZU	So there's always the more conservative method.
rLlZpnT02ZU	The first one, the only thing you're losing
rLlZpnT02ZU	is the rate of convergence of the central limit theorem.
rLlZpnT02ZU	So if n is large enough so that the central limit theorem
rLlZpnT02ZU	approximation is very good, then that's all you're
rLlZpnT02ZU	going to be losing.
rLlZpnT02ZU	Of course, the price you pay is that your confidence interval
rLlZpnT02ZU	is wider than it would be if you were
rLlZpnT02ZU	to use Slutsky for this particular problem,
rLlZpnT02ZU	typically wider.
rLlZpnT02ZU	Actually, it is always wider, because Xn bar--
rLlZpnT02ZU	1 minus Xn bar is always less than 1/4 as well.
rLlZpnT02ZU	And so that's the first thing you--
rLlZpnT02ZU	so Slutsky basically adds your relying on the central limit--
rLlZpnT02ZU	your relying on the asymptotics again.
rLlZpnT02ZU	Now of course, you don't want to be conservative,
rLlZpnT02ZU	because you actually want to squeeze as much from your data
rLlZpnT02ZU	as you can.
rLlZpnT02ZU	So it depends on how comfortable and how critical it is for you
rLlZpnT02ZU	to put valid error bars.
rLlZpnT02ZU	If they're valid in the asymptotics,
rLlZpnT02ZU	then maybe you're actually going to go with Slutsky
rLlZpnT02ZU	so it actually gives you slightly narrower confidence
rLlZpnT02ZU	intervals and so you feel like you're a little more--
rLlZpnT02ZU	you have a more precise answer.
rLlZpnT02ZU	Now, if you really need to be super-conservative,
rLlZpnT02ZU	then you're actually going to go with the P1 minus P.
rLlZpnT02ZU	Actually, if you need to be even more conservative,
rLlZpnT02ZU	you are going to go with Hoeffding's so you don't even
rLlZpnT02ZU	have to rely on the asymptotics level at all.
rLlZpnT02ZU	But then you're confidence interval
rLlZpnT02ZU	becomes twice as wide and twice as wide
rLlZpnT02ZU	and it becomes wider and wider as you go.
rLlZpnT02ZU	So depends on--
rLlZpnT02ZU	I mean, there's a lot of data in statistics
rLlZpnT02ZU	which is gauging how critical it is for you to output
rLlZpnT02ZU	valid error bounds or if they're really just here
rLlZpnT02ZU	to be indicative of the precision of the estimator you
rLlZpnT02ZU	gave from a more qualitative perspective.
rLlZpnT02ZU	AUDIENCE: So the error there is [INAUDIBLE]??
rLlZpnT02ZU	PHILIPPE RIGOLLET: Yeah.
rLlZpnT02ZU	So here, there's basically a bunch of errors.
rLlZpnT02ZU	There's one that's-- so there's a theorem called Berry-Esseen
rLlZpnT02ZU	that quantifies how far this probability is from 1 minus
rLlZpnT02ZU	alpha, but the constants are terrible.
rLlZpnT02ZU	So it's not very helpful, but it tells you
rLlZpnT02ZU	as n grows how smaller this thing grows--
rLlZpnT02ZU	becomes smaller.
rLlZpnT02ZU	And then for Slutsky, again you're
rLlZpnT02ZU	multiplying something that converges by something that
rLlZpnT02ZU	fluctuates around 1, so you need to understand
rLlZpnT02ZU	how this thing fluctuates.
rLlZpnT02ZU	Now, there's something that shows up.
rLlZpnT02ZU	Basically, what is the slope of the function 1
rLlZpnT02ZU	over square root of X1 minus X around the value
rLlZpnT02ZU	you're interested in?
rLlZpnT02ZU	And so if this function is super-sharp,
rLlZpnT02ZU	then small fluctuations of Xn bar around this expectation
rLlZpnT02ZU	are going to lead to really high fluctuations
rLlZpnT02ZU	of the function itself.
rLlZpnT02ZU	So if you're looking at--
rLlZpnT02ZU	if you have f of Xn bar and f around say the true P,
rLlZpnT02ZU	if f is really sharp like that, then
rLlZpnT02ZU	if you move a little bit here, then you're
rLlZpnT02ZU	going to move really a lot on the y-axis.
rLlZpnT02ZU	So that's what the function here-- the function
rLlZpnT02ZU	you're interested in is 1 over square root of X1 minus X.
rLlZpnT02ZU	So what does this function look like around the point where you
rLlZpnT02ZU	think P is the true parameter?
rLlZpnT02ZU	Its derivative really is what matters.
rLlZpnT02ZU	OK?
rLlZpnT02ZU	Any other question.
rLlZpnT02ZU	OK.
rLlZpnT02ZU	So it's important, because now we're
rLlZpnT02ZU	going to switch to the real let's do some hardcore
rLlZpnT02ZU	computation type of things.
rLlZpnT02ZU	All right.

rLlZpnT02ZU

PHILIPPE RIGOLLET: So that's not Slutsky, right?

rLlZpnT02ZU

AUDIENCE: That's [INAUDIBLE].

rLlZpnT02ZU

PHILIPPE RIGOLLET: So Slutsky tells you that if you--

rLlZpnT02ZU

Slutsky's about combining two types of convergence.

rLlZpnT02ZU

So Slutsky tells you that if you actually

rLlZpnT02ZU

have one Xn that converges to X in distribution and Yn

rLlZpnT02ZU

that converges to Y in probability, then

rLlZpnT02ZU

you can actually multiply Xn and Yn

rLlZpnT02ZU

and get that the limit in distribution

rLlZpnT02ZU

is the product of X and Y, where X is now a constant.

rLlZpnT02ZU

And here we have the constant, which is 1.

rLlZpnT02ZU

But I did that already, right?

rLlZpnT02ZU

Using Slutsky to replace it for the--

rLlZpnT02ZU

to replace P by Xn bar, we've done

rLlZpnT02ZU

that last time, maybe a couple of times ago, actually.

rLlZpnT02ZU

Yeah.

rLlZpnT02ZU

AUDIENCE: So I guess these statements are [INAUDIBLE]..

rLlZpnT02ZU

PHILIPPE RIGOLLET: That's correct.

rLlZpnT02ZU

AUDIENCE: So could we like figure out [INAUDIBLE]

rLlZpnT02ZU

can we set a finite [INAUDIBLE].

rLlZpnT02ZU

PHILIPPE RIGOLLET: So of course, the short answer is no.

rLlZpnT02ZU

So here's how you would go about thinking

rLlZpnT02ZU

about which method is better.

rLlZpnT02ZU

So there's always the more conservative method.

rLlZpnT02ZU

The first one, the only thing you're losing

rLlZpnT02ZU

is the rate of convergence of the central limit theorem.

rLlZpnT02ZU

So if n is large enough so that the central limit theorem

rLlZpnT02ZU

approximation is very good, then that's all you're

rLlZpnT02ZU

going to be losing.

rLlZpnT02ZU

Of course, the price you pay is that your confidence interval

rLlZpnT02ZU

is wider than it would be if you were

rLlZpnT02ZU

to use Slutsky for this particular problem,

rLlZpnT02ZU

typically wider.

rLlZpnT02ZU

Actually, it is always wider, because Xn bar--

rLlZpnT02ZU

1 minus Xn bar is always less than 1/4 as well.

rLlZpnT02ZU

And so that's the first thing you--

rLlZpnT02ZU

so Slutsky basically adds your relying on the central limit--

rLlZpnT02ZU

your relying on the asymptotics again.

rLlZpnT02ZU

Now of course, you don't want to be conservative,

rLlZpnT02ZU

because you actually want to squeeze as much from your data

rLlZpnT02ZU

as you can.

rLlZpnT02ZU

So it depends on how comfortable and how critical it is for you

rLlZpnT02ZU

to put valid error bars.

rLlZpnT02ZU

If they're valid in the asymptotics,

rLlZpnT02ZU

then maybe you're actually going to go with Slutsky

rLlZpnT02ZU

so it actually gives you slightly narrower confidence

rLlZpnT02ZU

intervals and so you feel like you're a little more--

rLlZpnT02ZU

you have a more precise answer.

rLlZpnT02ZU

Now, if you really need to be super-conservative,

rLlZpnT02ZU

then you're actually going to go with the P1 minus P.

rLlZpnT02ZU

Actually, if you need to be even more conservative,

rLlZpnT02ZU

you are going to go with Hoeffding's so you don't even

rLlZpnT02ZU

have to rely on the asymptotics level at all.

rLlZpnT02ZU

But then you're confidence interval

rLlZpnT02ZU

becomes twice as wide and twice as wide

rLlZpnT02ZU

and it becomes wider and wider as you go.

rLlZpnT02ZU

So depends on--

rLlZpnT02ZU

I mean, there's a lot of data in statistics

rLlZpnT02ZU

which is gauging how critical it is for you to output

rLlZpnT02ZU

valid error bounds or if they're really just here

rLlZpnT02ZU

to be indicative of the precision of the estimator you

rLlZpnT02ZU

gave from a more qualitative perspective.

rLlZpnT02ZU

AUDIENCE: So the error there is [INAUDIBLE]??

rLlZpnT02ZU

PHILIPPE RIGOLLET: Yeah.

rLlZpnT02ZU

So here, there's basically a bunch of errors.

rLlZpnT02ZU

There's one that's-- so there's a theorem called Berry-Esseen

rLlZpnT02ZU

that quantifies how far this probability is from 1 minus

rLlZpnT02ZU

alpha, but the constants are terrible.

rLlZpnT02ZU

So it's not very helpful, but it tells you

rLlZpnT02ZU

as n grows how smaller this thing grows--

rLlZpnT02ZU

becomes smaller.

rLlZpnT02ZU

And then for Slutsky, again you're

rLlZpnT02ZU

multiplying something that converges by something that

rLlZpnT02ZU

fluctuates around 1, so you need to understand

rLlZpnT02ZU

how this thing fluctuates.

rLlZpnT02ZU

Now, there's something that shows up.

rLlZpnT02ZU

Basically, what is the slope of the function 1

rLlZpnT02ZU

over square root of X1 minus X around the value

rLlZpnT02ZU

you're interested in?

rLlZpnT02ZU

And so if this function is super-sharp,

rLlZpnT02ZU

then small fluctuations of Xn bar around this expectation

rLlZpnT02ZU

are going to lead to really high fluctuations

rLlZpnT02ZU

of the function itself.

rLlZpnT02ZU

So if you're looking at--

rLlZpnT02ZU

if you have f of Xn bar and f around say the true P,

rLlZpnT02ZU

if f is really sharp like that, then

rLlZpnT02ZU

if you move a little bit here, then you're

rLlZpnT02ZU

going to move really a lot on the y-axis.

rLlZpnT02ZU

So that's what the function here-- the function

rLlZpnT02ZU

you're interested in is 1 over square root of X1 minus X.

rLlZpnT02ZU

So what does this function look like around the point where you

rLlZpnT02ZU

think P is the true parameter?

rLlZpnT02ZU

Its derivative really is what matters.

rLlZpnT02ZU

OK?

rLlZpnT02ZU

Any other question.

rLlZpnT02ZU

OK.

rLlZpnT02ZU

So it's important, because now we're

rLlZpnT02ZU

going to switch to the real let's do some hardcore

rLlZpnT02ZU

computation type of things.

rLlZpnT02ZU

All right.