jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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rLlZpnT02ZU	if you do it for each simple event, it's little x.
rLlZpnT02ZU	It's actually simple.
rLlZpnT02ZU	Now, if we have continuous random variables-- so
rLlZpnT02ZU	by the way, I didn't mention, but discrete means Bernoulli.
rLlZpnT02ZU	Binomial, but not only those that have finite support,
rLlZpnT02ZU	like Bernoulli has support of size 2,
rLlZpnT02ZU	binomial NP has support of size n--
rLlZpnT02ZU	there's n possible values it can take-- but also Poisson.
rLlZpnT02ZU	Poisson distribution can take an infinite number
rLlZpnT02ZU	of values, all the positive integers,
rLlZpnT02ZU	non-negative integers.
rLlZpnT02ZU	And so now we have also the continuous ones,
rLlZpnT02ZU	such as Gaussian, exponential.
rLlZpnT02ZU	And what characterizes those guys is that they
rLlZpnT02ZU	have a probability density.
rLlZpnT02ZU	So the density, remember the way I
rLlZpnT02ZU	use my density is when I want to compute
rLlZpnT02ZU	the probability of belonging to some event A.
rLlZpnT02ZU	The probability of X falling to some subset of the real line A
rLlZpnT02ZU	is simply the integral of the density on this set.
rLlZpnT02ZU	That's the famous area under the curve thing.
rLlZpnT02ZU	So since for each possible value, the probability at X--
rLlZpnT02ZU	so I hope you remember that stuff.
rLlZpnT02ZU	That's just probably something that you
rLlZpnT02ZU	must remember from probability.
rLlZpnT02ZU	But essentially, we know that the probability that X is equal
rLlZpnT02ZU	to little x is 0 for a continuous random variable,
rLlZpnT02ZU	for all possible X's.
rLlZpnT02ZU	There's just none of them that actually gets weight.
rLlZpnT02ZU	So what we have to do is to describe the fact that it's
rLlZpnT02ZU	in some little region.
rLlZpnT02ZU	So the probability that it's in some interval, say, a, b, this
rLlZpnT02ZU	is the integral between A and B of f theta of X, dx.
rLlZpnT02ZU	So I have this density, such as the Gaussian one.
rLlZpnT02ZU	And the probability that I belong to the interval a,
rLlZpnT02ZU	b is just the area under the curve between A and B.
rLlZpnT02ZU	If you don't remember that, please take immediate remedy.
rLlZpnT02ZU	So this function f, just like P, is non-negative.
rLlZpnT02ZU	And rather than summing to 1, it integrates to 1
rLlZpnT02ZU	when I integrate it over the entire sample space E.
rLlZpnT02ZU	And now the total variation, well, it
rLlZpnT02ZU	takes basically the same form.
rLlZpnT02ZU	I said that you essentially replace sums
rLlZpnT02ZU	by integrals when you're dealing with densities.
rLlZpnT02ZU	And here, it's just saying, rather than having
rLlZpnT02ZU	1/2 of the sum of the absolute values,
rLlZpnT02ZU	you have 1/2 of the integral of the absolute value
rLlZpnT02ZU	of the difference.
rLlZpnT02ZU	Again, if I give you two densities
rLlZpnT02ZU	and if you're not too bad at calculus, which you will often
rLlZpnT02ZU	be, because there's lots of them you can actually not compute.
rLlZpnT02ZU	But if I gave you, for example, two Gaussian densities,
rLlZpnT02ZU	exponential minus x squared, blah, blah, blah, and I say,
rLlZpnT02ZU	just compute the total variation distance,
rLlZpnT02ZU	you could actually write it as an integral.
rLlZpnT02ZU	Now, whether you can actually reduce this integral
rLlZpnT02ZU	to some particular number is another story.
rLlZpnT02ZU	But you could technically do it.
rLlZpnT02ZU	So now, you have actually a handle on this thing
rLlZpnT02ZU	and you could technically ask Mathematica,
rLlZpnT02ZU	whereas asking Mathematica to take
rLlZpnT02ZU	the max over all possible events is going to be difficult.
rLlZpnT02ZU	All right.
rLlZpnT02ZU	So the total variation has some properties.
rLlZpnT02ZU	So let's keep on the board the definition that
rLlZpnT02ZU	involves, say, the densities.
rLlZpnT02ZU	So think Gaussian in your mind.
rLlZpnT02ZU	And you have two Gaussians, one with mean theta
rLlZpnT02ZU	and one with mean theta prime.
rLlZpnT02ZU	And I'm looking at the total variation between those two
rLlZpnT02ZU	guys.
rLlZpnT02ZU	So if I look at P theta minus--
rLlZpnT02ZU	sorry.
rLlZpnT02ZU	TV between P theta and P theta prime, this
rLlZpnT02ZU	is equal to 1/2 of the integral between f theta, f theta prime.
rLlZpnT02ZU	And when I don't write it--
rLlZpnT02ZU	so I don't write the X, dx but it's there.
rLlZpnT02ZU	And then I integrate over E.
rLlZpnT02ZU	So what is this thing doing for me?
rLlZpnT02ZU	It's just saying, well, if I have-- so
rLlZpnT02ZU	think of two Gaussians.
rLlZpnT02ZU	For example, I have one that's here and one that's here.
rLlZpnT02ZU	So this is let's say f theta, f theta prime.
rLlZpnT02ZU	This guy is doing what?
rLlZpnT02ZU	It's computing the absolute value of the difference
rLlZpnT02ZU	between f and f theta prime.
rLlZpnT02ZU	You can check for yourself that graphically, this I
rLlZpnT02ZU	can represent as an area not under the curve,
rLlZpnT02ZU	but between the curves.
rLlZpnT02ZU	So this is this guy.
rLlZpnT02ZU	Now, this guy is really the integral of the absolute value.
rLlZpnT02ZU	So this thing here, this area, this
rLlZpnT02ZU	is 2 times the total variation.
rLlZpnT02ZU	The scaling 1/2 really doesn't matter.
rLlZpnT02ZU	It's just if I want to have an actual correspondence
rLlZpnT02ZU	between the maximum and the other guy, I have to do this.
rLlZpnT02ZU	So this is what it looks like.
rLlZpnT02ZU	So we have this definition.
rLlZpnT02ZU	And so we have a couple of properties that come into this.
rLlZpnT02ZU	The first one is that it's symmetric.

rLlZpnT02ZU

if you do it for each simple event, it's little x.

rLlZpnT02ZU

It's actually simple.

rLlZpnT02ZU

Now, if we have continuous random variables-- so

rLlZpnT02ZU

by the way, I didn't mention, but discrete means Bernoulli.

rLlZpnT02ZU

Binomial, but not only those that have finite support,

rLlZpnT02ZU

like Bernoulli has support of size 2,

rLlZpnT02ZU

binomial NP has support of size n--

rLlZpnT02ZU

there's n possible values it can take-- but also Poisson.

rLlZpnT02ZU

Poisson distribution can take an infinite number

rLlZpnT02ZU

of values, all the positive integers,

rLlZpnT02ZU

non-negative integers.

rLlZpnT02ZU

And so now we have also the continuous ones,

rLlZpnT02ZU

such as Gaussian, exponential.

rLlZpnT02ZU

And what characterizes those guys is that they

rLlZpnT02ZU

have a probability density.

rLlZpnT02ZU

So the density, remember the way I

rLlZpnT02ZU

use my density is when I want to compute

rLlZpnT02ZU

the probability of belonging to some event A.

rLlZpnT02ZU

The probability of X falling to some subset of the real line A

rLlZpnT02ZU

is simply the integral of the density on this set.

rLlZpnT02ZU

That's the famous area under the curve thing.

rLlZpnT02ZU

So since for each possible value, the probability at X--

rLlZpnT02ZU

so I hope you remember that stuff.

rLlZpnT02ZU

That's just probably something that you

rLlZpnT02ZU

must remember from probability.

rLlZpnT02ZU

But essentially, we know that the probability that X is equal

rLlZpnT02ZU

to little x is 0 for a continuous random variable,

rLlZpnT02ZU

for all possible X's.

rLlZpnT02ZU

There's just none of them that actually gets weight.

rLlZpnT02ZU

So what we have to do is to describe the fact that it's

rLlZpnT02ZU

in some little region.

rLlZpnT02ZU

So the probability that it's in some interval, say, a, b, this

rLlZpnT02ZU

is the integral between A and B of f theta of X, dx.

rLlZpnT02ZU

So I have this density, such as the Gaussian one.

rLlZpnT02ZU

And the probability that I belong to the interval a,

rLlZpnT02ZU

b is just the area under the curve between A and B.

rLlZpnT02ZU

If you don't remember that, please take immediate remedy.

rLlZpnT02ZU

So this function f, just like P, is non-negative.

rLlZpnT02ZU

And rather than summing to 1, it integrates to 1

rLlZpnT02ZU

when I integrate it over the entire sample space E.

rLlZpnT02ZU

And now the total variation, well, it

rLlZpnT02ZU

takes basically the same form.

rLlZpnT02ZU

I said that you essentially replace sums

rLlZpnT02ZU

by integrals when you're dealing with densities.

rLlZpnT02ZU

And here, it's just saying, rather than having

rLlZpnT02ZU

1/2 of the sum of the absolute values,

rLlZpnT02ZU

you have 1/2 of the integral of the absolute value

rLlZpnT02ZU

of the difference.

rLlZpnT02ZU

Again, if I give you two densities

rLlZpnT02ZU

and if you're not too bad at calculus, which you will often

rLlZpnT02ZU

be, because there's lots of them you can actually not compute.

rLlZpnT02ZU

But if I gave you, for example, two Gaussian densities,

rLlZpnT02ZU

exponential minus x squared, blah, blah, blah, and I say,

rLlZpnT02ZU

just compute the total variation distance,

rLlZpnT02ZU

you could actually write it as an integral.

rLlZpnT02ZU

Now, whether you can actually reduce this integral

rLlZpnT02ZU

to some particular number is another story.

rLlZpnT02ZU

But you could technically do it.

rLlZpnT02ZU

So now, you have actually a handle on this thing

rLlZpnT02ZU

and you could technically ask Mathematica,

rLlZpnT02ZU

whereas asking Mathematica to take

rLlZpnT02ZU

the max over all possible events is going to be difficult.

rLlZpnT02ZU

All right.

rLlZpnT02ZU

So the total variation has some properties.

rLlZpnT02ZU

So let's keep on the board the definition that

rLlZpnT02ZU

involves, say, the densities.

rLlZpnT02ZU

So think Gaussian in your mind.

rLlZpnT02ZU

And you have two Gaussians, one with mean theta

rLlZpnT02ZU

and one with mean theta prime.

rLlZpnT02ZU

And I'm looking at the total variation between those two

rLlZpnT02ZU

guys.

rLlZpnT02ZU

So if I look at P theta minus--

rLlZpnT02ZU

sorry.

rLlZpnT02ZU

TV between P theta and P theta prime, this

rLlZpnT02ZU

is equal to 1/2 of the integral between f theta, f theta prime.

rLlZpnT02ZU

And when I don't write it--

rLlZpnT02ZU

so I don't write the X, dx but it's there.

rLlZpnT02ZU

And then I integrate over E.

rLlZpnT02ZU

So what is this thing doing for me?

rLlZpnT02ZU

It's just saying, well, if I have-- so

rLlZpnT02ZU

think of two Gaussians.

rLlZpnT02ZU

For example, I have one that's here and one that's here.

rLlZpnT02ZU

So this is let's say f theta, f theta prime.

rLlZpnT02ZU

This guy is doing what?

rLlZpnT02ZU

It's computing the absolute value of the difference

rLlZpnT02ZU

between f and f theta prime.

rLlZpnT02ZU

You can check for yourself that graphically, this I

rLlZpnT02ZU

can represent as an area not under the curve,

rLlZpnT02ZU

but between the curves.

rLlZpnT02ZU

So this is this guy.

rLlZpnT02ZU

Now, this guy is really the integral of the absolute value.

rLlZpnT02ZU

So this thing here, this area, this

rLlZpnT02ZU

is 2 times the total variation.

rLlZpnT02ZU

The scaling 1/2 really doesn't matter.

rLlZpnT02ZU

It's just if I want to have an actual correspondence

rLlZpnT02ZU

between the maximum and the other guy, I have to do this.

rLlZpnT02ZU

So this is what it looks like.

rLlZpnT02ZU

So we have this definition.

rLlZpnT02ZU

And so we have a couple of properties that come into this.

rLlZpnT02ZU

The first one is that it's symmetric.