jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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rLlZpnT02ZU	So just let's look at back into our favorite example, which
rLlZpnT02ZU	is the average of Bernoulli random variables,
rLlZpnT02ZU	so we studied that maybe that's the third time already.
rLlZpnT02ZU	So the sample average, Xn bar, is a strongly consistent
rLlZpnT02ZU	estimator of p.
rLlZpnT02ZU	That was one of the properties that we wanted.
rLlZpnT02ZU	Strongly consistent means that as n goes to infinity,
rLlZpnT02ZU	it converges almost surely to the true parameter.
rLlZpnT02ZU	That's the strong law of large number.
rLlZpnT02ZU	It is consistent also, because it's strongly consistent,
rLlZpnT02ZU	so it also converges in probability,
rLlZpnT02ZU	which makes it consistent.
rLlZpnT02ZU	It's unbiased.
rLlZpnT02ZU	We've seen that.
rLlZpnT02ZU	We've actually computed its quadratic risk.
rLlZpnT02ZU	And now what I have is that if I look at--
rLlZpnT02ZU	thanks to the central limit theorem, we actually did this.
rLlZpnT02ZU	We built a confidence interval at level 1 minus alpha--
rLlZpnT02ZU	asymptotic level, sorry, asymptotic level 1 minus alpha.
rLlZpnT02ZU	And so here, this is how we did it.
rLlZpnT02ZU	Let me just go through it again.
rLlZpnT02ZU	So we know from the central limit theorem--
rLlZpnT02ZU	so the central limit theorem tells us
rLlZpnT02ZU	that Xn bar minus p divided by square root of p1 minus p,
rLlZpnT02ZU	square root of n converges in distribution as n
rLlZpnT02ZU	goes to infinity to some standard normal distribution.
rLlZpnT02ZU	So what it means is that if I look at the probability
rLlZpnT02ZU	under the true p, that's square root of n, Xn bar
rLlZpnT02ZU	minus p divided by square root of p1 minus p,
rLlZpnT02ZU	it's less than Q alpha over 2, where this is
rLlZpnT02ZU	the definition of the quintile.
rLlZpnT02ZU	Then this guy-- and I'm actually going to use the same notation,
rLlZpnT02ZU	limit as n goes to infinity, this is the same thing.
rLlZpnT02ZU	So this is actually going to be equal to 1 minus alpha.
rLlZpnT02ZU	That's exactly what I did last time.
rLlZpnT02ZU	This is by definition of the quintile of a standard Gaussian
rLlZpnT02ZU	and of a limit in distribution.
rLlZpnT02ZU	So the probabilities computed on this guy in the limit converges
rLlZpnT02ZU	to the probability computed on this guy.
rLlZpnT02ZU	And we know that this is just the probability
rLlZpnT02ZU	that the absolute value of sum n 0, 1
rLlZpnT02ZU	is less than Q alpha over 2.
rLlZpnT02ZU	And so in particular, if it's equal,
rLlZpnT02ZU	then I can put some larger than or equal to,
rLlZpnT02ZU	which guarantees my asymptotic confidence level.
rLlZpnT02ZU	And I just solve for p.
rLlZpnT02ZU	So this is equivalent to the limit
rLlZpnT02ZU	as n goes to infinity of the probability
rLlZpnT02ZU	that theta is between Xn bar minus Q
rLlZpnT02ZU	alpha over 2 divided by--
rLlZpnT02ZU	times square root of p1 minus p divided by square root of n, Xn
rLlZpnT02ZU	bar plus q alpha over 2, square root of p1 minus p
rLlZpnT02ZU	divided by square root of n is larger than or equal
rLlZpnT02ZU	to 1 minus alpha.
rLlZpnT02ZU	And so there you go.
rLlZpnT02ZU	I have my confidence interval.
rLlZpnT02ZU	Except that's not, right?
rLlZpnT02ZU	We just said that the bounds of a confidence interval
rLlZpnT02ZU	may not depend on the unknown parameter.
rLlZpnT02ZU	And here, they do.
rLlZpnT02ZU	And so we actually came up with two ways
rLlZpnT02ZU	of getting rid of this.
rLlZpnT02ZU	Since we only need this thing-- so this thing, as we said,
rLlZpnT02ZU	is really equal.
rLlZpnT02ZU	Every time I'm going to make this guy smaller
rLlZpnT02ZU	and this guy larger, I'm only going
rLlZpnT02ZU	to increase the probability.
rLlZpnT02ZU	And so what we do is we actually just take
rLlZpnT02ZU	the largest possible value for p1 minus
rLlZpnT02ZU	p, which makes the interval as large as possible.
rLlZpnT02ZU	And so now I have this.
rLlZpnT02ZU	I just do one of the two tricks.
rLlZpnT02ZU	I replace p1 minus p by their upper bound, which is 1/4.
rLlZpnT02ZU	As we said, p1 minus p, the function looks like this.
rLlZpnT02ZU	So I just take the value here at 1/2.
rLlZpnT02ZU	Or, I can use Slutsky and say that if I replace p by Xn bar,
rLlZpnT02ZU	that's the same as just replacing p by Xn bar here.
rLlZpnT02ZU	And by Slutsky, we know that this is actually converging
rLlZpnT02ZU	also to some standard Gaussian.
rLlZpnT02ZU	We've seen that when we saw Slutsky as an example.
rLlZpnT02ZU	And so those two things-- actually,
rLlZpnT02ZU	just because I'm taking the limit
rLlZpnT02ZU	and I'm only caring about the asymptotic confidence level,
rLlZpnT02ZU	I can actually just plug in consistent quantities in there,
rLlZpnT02ZU	such as Xn bar where I don't have a p.
rLlZpnT02ZU	And that gives me another confidence interval.
rLlZpnT02ZU	All right.
rLlZpnT02ZU	So this by now, hopefully after doing it three times,
rLlZpnT02ZU	you should really, really be comfortable with just creating
rLlZpnT02ZU	this confidence interval.
rLlZpnT02ZU	We did it three times in class.
rLlZpnT02ZU	I think you probably did it another couple times
rLlZpnT02ZU	in your homework.
rLlZpnT02ZU	So just make sure you're comfortable with this.
rLlZpnT02ZU	All right.
rLlZpnT02ZU	That's one of the basic things you would want to know.
rLlZpnT02ZU	Are there any questions?
rLlZpnT02ZU	Yes.
rLlZpnT02ZU	AUDIENCE: So Slutsky holds for any single response set p.
rLlZpnT02ZU	But Xn converges [INAUDIBLE].

rLlZpnT02ZU

So just let's look at back into our favorite example, which

rLlZpnT02ZU

is the average of Bernoulli random variables,

rLlZpnT02ZU

so we studied that maybe that's the third time already.

rLlZpnT02ZU

So the sample average, Xn bar, is a strongly consistent

rLlZpnT02ZU

estimator of p.

rLlZpnT02ZU

That was one of the properties that we wanted.

rLlZpnT02ZU

Strongly consistent means that as n goes to infinity,

rLlZpnT02ZU

it converges almost surely to the true parameter.

rLlZpnT02ZU

That's the strong law of large number.

rLlZpnT02ZU

It is consistent also, because it's strongly consistent,

rLlZpnT02ZU

so it also converges in probability,

rLlZpnT02ZU

which makes it consistent.

rLlZpnT02ZU

It's unbiased.

rLlZpnT02ZU

We've seen that.

rLlZpnT02ZU

We've actually computed its quadratic risk.

rLlZpnT02ZU

And now what I have is that if I look at--

rLlZpnT02ZU

thanks to the central limit theorem, we actually did this.

rLlZpnT02ZU

We built a confidence interval at level 1 minus alpha--

rLlZpnT02ZU

asymptotic level, sorry, asymptotic level 1 minus alpha.

rLlZpnT02ZU

And so here, this is how we did it.

rLlZpnT02ZU

Let me just go through it again.

rLlZpnT02ZU

So we know from the central limit theorem--

rLlZpnT02ZU

so the central limit theorem tells us

rLlZpnT02ZU

that Xn bar minus p divided by square root of p1 minus p,

rLlZpnT02ZU

square root of n converges in distribution as n

rLlZpnT02ZU

goes to infinity to some standard normal distribution.

rLlZpnT02ZU

So what it means is that if I look at the probability

rLlZpnT02ZU

under the true p, that's square root of n, Xn bar

rLlZpnT02ZU

minus p divided by square root of p1 minus p,

rLlZpnT02ZU

it's less than Q alpha over 2, where this is

rLlZpnT02ZU

the definition of the quintile.

rLlZpnT02ZU

Then this guy-- and I'm actually going to use the same notation,

rLlZpnT02ZU

limit as n goes to infinity, this is the same thing.

rLlZpnT02ZU

So this is actually going to be equal to 1 minus alpha.

rLlZpnT02ZU

That's exactly what I did last time.

rLlZpnT02ZU

This is by definition of the quintile of a standard Gaussian

rLlZpnT02ZU

and of a limit in distribution.

rLlZpnT02ZU

So the probabilities computed on this guy in the limit converges

rLlZpnT02ZU

to the probability computed on this guy.

rLlZpnT02ZU

And we know that this is just the probability

rLlZpnT02ZU

that the absolute value of sum n 0, 1

rLlZpnT02ZU

is less than Q alpha over 2.

rLlZpnT02ZU

And so in particular, if it's equal,

rLlZpnT02ZU

then I can put some larger than or equal to,

rLlZpnT02ZU

which guarantees my asymptotic confidence level.

rLlZpnT02ZU

And I just solve for p.

rLlZpnT02ZU

So this is equivalent to the limit

rLlZpnT02ZU

as n goes to infinity of the probability

rLlZpnT02ZU

that theta is between Xn bar minus Q

rLlZpnT02ZU

alpha over 2 divided by--

rLlZpnT02ZU

times square root of p1 minus p divided by square root of n, Xn

rLlZpnT02ZU

bar plus q alpha over 2, square root of p1 minus p

rLlZpnT02ZU

divided by square root of n is larger than or equal

rLlZpnT02ZU

to 1 minus alpha.

rLlZpnT02ZU

And so there you go.

rLlZpnT02ZU

I have my confidence interval.

rLlZpnT02ZU

Except that's not, right?

rLlZpnT02ZU

We just said that the bounds of a confidence interval

rLlZpnT02ZU

may not depend on the unknown parameter.

rLlZpnT02ZU

And here, they do.

rLlZpnT02ZU

And so we actually came up with two ways

rLlZpnT02ZU

of getting rid of this.

rLlZpnT02ZU

Since we only need this thing-- so this thing, as we said,

rLlZpnT02ZU

is really equal.

rLlZpnT02ZU

Every time I'm going to make this guy smaller

rLlZpnT02ZU

and this guy larger, I'm only going

rLlZpnT02ZU

to increase the probability.

rLlZpnT02ZU

And so what we do is we actually just take

rLlZpnT02ZU

the largest possible value for p1 minus

rLlZpnT02ZU

p, which makes the interval as large as possible.

rLlZpnT02ZU

And so now I have this.

rLlZpnT02ZU

I just do one of the two tricks.

rLlZpnT02ZU

I replace p1 minus p by their upper bound, which is 1/4.

rLlZpnT02ZU

As we said, p1 minus p, the function looks like this.

rLlZpnT02ZU

So I just take the value here at 1/2.

rLlZpnT02ZU

Or, I can use Slutsky and say that if I replace p by Xn bar,

rLlZpnT02ZU

that's the same as just replacing p by Xn bar here.

rLlZpnT02ZU

And by Slutsky, we know that this is actually converging

rLlZpnT02ZU

also to some standard Gaussian.

rLlZpnT02ZU

We've seen that when we saw Slutsky as an example.

rLlZpnT02ZU

And so those two things-- actually,

rLlZpnT02ZU

just because I'm taking the limit

rLlZpnT02ZU

and I'm only caring about the asymptotic confidence level,

rLlZpnT02ZU

I can actually just plug in consistent quantities in there,

rLlZpnT02ZU

such as Xn bar where I don't have a p.

rLlZpnT02ZU

And that gives me another confidence interval.

rLlZpnT02ZU

All right.

rLlZpnT02ZU

So this by now, hopefully after doing it three times,

rLlZpnT02ZU

you should really, really be comfortable with just creating

rLlZpnT02ZU

this confidence interval.

rLlZpnT02ZU

We did it three times in class.

rLlZpnT02ZU

I think you probably did it another couple times

rLlZpnT02ZU

in your homework.

rLlZpnT02ZU

So just make sure you're comfortable with this.

rLlZpnT02ZU

All right.

rLlZpnT02ZU

That's one of the basic things you would want to know.

rLlZpnT02ZU

Are there any questions?

rLlZpnT02ZU

Yes.

rLlZpnT02ZU

AUDIENCE: So Slutsky holds for any single response set p.

rLlZpnT02ZU

But Xn converges [INAUDIBLE].