jeongyoun/VideoTranscript_Math_NER · Datasets at Hugging Face

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rLlZpnT02ZU	that I get different samples every time
rLlZpnT02ZU	should somewhat vanish.
rLlZpnT02ZU	And so what I want is to have a small bias, hopefully a 0 bias.
rLlZpnT02ZU	If this thing is 0, then we see that the estimator is unbiased.
rLlZpnT02ZU	So this is definitely a property that we
rLlZpnT02ZU	are going to be looking for in an estimator,
rLlZpnT02ZU	trying to find them to be unbiased.
rLlZpnT02ZU	But we'll see that it's actually maybe not enough.
rLlZpnT02ZU	So unbiasedness should not be something
rLlZpnT02ZU	you lose your sleep over.
rLlZpnT02ZU	Something that's slightly better is the risk, really
rLlZpnT02ZU	the quadratics risk, which is expectation of--
rLlZpnT02ZU	so if I have an estimator, theta hat,
rLlZpnT02ZU	I'm going to look at the expectation of theta hat n
rLlZpnT02ZU	minus theta squared.
rLlZpnT02ZU	And what we showed last time is that we can actually--
rLlZpnT02ZU	by inserting in there, adding and removing
rLlZpnT02ZU	the expectation of theta hat, we actually
rLlZpnT02ZU	get something where this thing can
rLlZpnT02ZU	be decomposed as the square of the bias plus the variance,
rLlZpnT02ZU	which is just the expectation of theta hat minus its expectation
rLlZpnT02ZU	squared.
rLlZpnT02ZU	That came from the fact that when
rLlZpnT02ZU	I added and removed the expectation of theta hat
rLlZpnT02ZU	in there, the cross-terms cancel.
rLlZpnT02ZU	All right.
rLlZpnT02ZU	So that was the bias squared, and this is the variance.
rLlZpnT02ZU	And so for example, if the quadratic risk goes to 0,
rLlZpnT02ZU	then that means that theta hat converges
rLlZpnT02ZU	to theta in the L2 sense.
rLlZpnT02ZU	And here we know that if we want this to go to 0,
rLlZpnT02ZU	since it's the sum of two positive terms,
rLlZpnT02ZU	we need to have both the bias that goes to 0
rLlZpnT02ZU	and the variance that goes to 0, so we
rLlZpnT02ZU	need to control both of those things.
rLlZpnT02ZU	And so there is usually an inherent trade-off
rLlZpnT02ZU	between getting a small bias and getting a small variance.
rLlZpnT02ZU	If you reduce one too much, then the variance of the other one
rLlZpnT02ZU	is going to--
rLlZpnT02ZU	then the other one is going to increase, or the opposite.
rLlZpnT02ZU	That happens a lot, but not so much, actually, in this class.
rLlZpnT02ZU	So let's just look at a couple of examples.
rLlZpnT02ZU	So am I planning--
rLlZpnT02ZU	yeah.
rLlZpnT02ZU	So examples.
rLlZpnT02ZU	So if I do, for example, X1, Xn, there are iid Bernoulli.
rLlZpnT02ZU	And I'm going to write it theta so
rLlZpnT02ZU	that we keep the same notation.
rLlZpnT02ZU	Then theta hat, what is the theta hat
rLlZpnT02ZU	that we proposed many times?
rLlZpnT02ZU	It's just X bar, Xn bar, the average of Xi's.
rLlZpnT02ZU	So what is the bias of this guy?
rLlZpnT02ZU	Well, to know the bias, I just have to remove theta
rLlZpnT02ZU	from the expectation.
rLlZpnT02ZU	What is the expectation of Xn bar?
rLlZpnT02ZU	Well, by linearity of the expectation,
rLlZpnT02ZU	it's just the average of the expectations.
rLlZpnT02ZU	But since all my Xi's are Bernouilli with the same theta,
rLlZpnT02ZU	then each of this guy is actually equal to theta.
rLlZpnT02ZU	So this thing is actually theta, which means
rLlZpnT02ZU	that this isn't biased, right?
rLlZpnT02ZU	Now, what is the variance of this guy?
rLlZpnT02ZU	So if you forgot the properties of the variance
rLlZpnT02ZU	for sum of independent random variables,
rLlZpnT02ZU	now it's time to wake up.
rLlZpnT02ZU	So we have the variance of something
rLlZpnT02ZU	that looks like 1 over n, the sum from i equal 1 to n of Xi.
rLlZpnT02ZU	So it's of the form variance of a constant times
rLlZpnT02ZU	a random variable.
rLlZpnT02ZU	So the first thing I'm going to do is pull out the constant.
rLlZpnT02ZU	But we know that the variance leaves on the square scale,
rLlZpnT02ZU	so when I pull out a constant outside of the variance,
rLlZpnT02ZU	it comes out with a square.
rLlZpnT02ZU	The variance of a times X is a-squared
rLlZpnT02ZU	times the variance of X, so this is equal to 1
rLlZpnT02ZU	over n squared times the variance of the sum.
rLlZpnT02ZU	So now we want to always do what we want to do.
rLlZpnT02ZU	So we have the variance of the sum.
rLlZpnT02ZU	We would like somehow to say that this
rLlZpnT02ZU	is the sum of the variances.
rLlZpnT02ZU	And in general, we are not allowed to say that,
rLlZpnT02ZU	but we are because my Xi's are actually independent.
rLlZpnT02ZU	So this is actually equal to 1 over n squared sum from i equal
rLlZpnT02ZU	1 to n of the variance of each of the Xi's.
rLlZpnT02ZU	And that's by independence, so this is basic probability.
rLlZpnT02ZU	And now, what is the variance of Xi's where again they're
rLlZpnT02ZU	all the same distribution, so the variance of Xi
rLlZpnT02ZU	is the same as the variance of X1.
rLlZpnT02ZU	And so each of those guys has variance what?
rLlZpnT02ZU	What is the variance of a Bernoulli?
rLlZpnT02ZU	We've said it once.
rLlZpnT02ZU	It's theta times 1 minus theta.
rLlZpnT02ZU	And so now I'm going to have the sum of n times a constant,
rLlZpnT02ZU	so I get n times the constant divided by n squared,
rLlZpnT02ZU	so one of the n's is going to cancel.
rLlZpnT02ZU	And so the whole thing here is actually
rLlZpnT02ZU	equal to theta, 1 minus theta divided by n.
rLlZpnT02ZU	So if I'm interested in the quadratic risk--
rLlZpnT02ZU	and again, I should just say risk,
rLlZpnT02ZU	because this is the only risk we're going

rLlZpnT02ZU

that I get different samples every time

rLlZpnT02ZU

should somewhat vanish.

rLlZpnT02ZU

And so what I want is to have a small bias, hopefully a 0 bias.

rLlZpnT02ZU

If this thing is 0, then we see that the estimator is unbiased.

rLlZpnT02ZU

So this is definitely a property that we

rLlZpnT02ZU

are going to be looking for in an estimator,

rLlZpnT02ZU

trying to find them to be unbiased.

rLlZpnT02ZU

But we'll see that it's actually maybe not enough.

rLlZpnT02ZU

So unbiasedness should not be something

rLlZpnT02ZU

you lose your sleep over.

rLlZpnT02ZU

Something that's slightly better is the risk, really

rLlZpnT02ZU

the quadratics risk, which is expectation of--

rLlZpnT02ZU

so if I have an estimator, theta hat,

rLlZpnT02ZU

I'm going to look at the expectation of theta hat n

rLlZpnT02ZU

minus theta squared.

rLlZpnT02ZU

And what we showed last time is that we can actually--

rLlZpnT02ZU

by inserting in there, adding and removing

rLlZpnT02ZU

the expectation of theta hat, we actually

rLlZpnT02ZU

get something where this thing can

rLlZpnT02ZU

be decomposed as the square of the bias plus the variance,

rLlZpnT02ZU

which is just the expectation of theta hat minus its expectation

rLlZpnT02ZU

squared.

rLlZpnT02ZU

That came from the fact that when

rLlZpnT02ZU

I added and removed the expectation of theta hat

rLlZpnT02ZU

in there, the cross-terms cancel.

rLlZpnT02ZU

All right.

rLlZpnT02ZU

So that was the bias squared, and this is the variance.

rLlZpnT02ZU

And so for example, if the quadratic risk goes to 0,

rLlZpnT02ZU

then that means that theta hat converges

rLlZpnT02ZU

to theta in the L2 sense.

rLlZpnT02ZU

And here we know that if we want this to go to 0,

rLlZpnT02ZU

since it's the sum of two positive terms,

rLlZpnT02ZU

we need to have both the bias that goes to 0

rLlZpnT02ZU

and the variance that goes to 0, so we

rLlZpnT02ZU

need to control both of those things.

rLlZpnT02ZU

And so there is usually an inherent trade-off

rLlZpnT02ZU

between getting a small bias and getting a small variance.

rLlZpnT02ZU

If you reduce one too much, then the variance of the other one

rLlZpnT02ZU

is going to--

rLlZpnT02ZU

then the other one is going to increase, or the opposite.

rLlZpnT02ZU

That happens a lot, but not so much, actually, in this class.

rLlZpnT02ZU

So let's just look at a couple of examples.

rLlZpnT02ZU

So am I planning--

rLlZpnT02ZU

yeah.

rLlZpnT02ZU

So examples.

rLlZpnT02ZU

So if I do, for example, X1, Xn, there are iid Bernoulli.

rLlZpnT02ZU

And I'm going to write it theta so

rLlZpnT02ZU

that we keep the same notation.

rLlZpnT02ZU

Then theta hat, what is the theta hat

rLlZpnT02ZU

that we proposed many times?

rLlZpnT02ZU

It's just X bar, Xn bar, the average of Xi's.

rLlZpnT02ZU

So what is the bias of this guy?

rLlZpnT02ZU

Well, to know the bias, I just have to remove theta

rLlZpnT02ZU

from the expectation.

rLlZpnT02ZU

What is the expectation of Xn bar?

rLlZpnT02ZU

Well, by linearity of the expectation,

rLlZpnT02ZU

it's just the average of the expectations.

rLlZpnT02ZU

But since all my Xi's are Bernouilli with the same theta,

rLlZpnT02ZU

then each of this guy is actually equal to theta.

rLlZpnT02ZU

So this thing is actually theta, which means

rLlZpnT02ZU

that this isn't biased, right?

rLlZpnT02ZU

Now, what is the variance of this guy?

rLlZpnT02ZU

So if you forgot the properties of the variance

rLlZpnT02ZU

for sum of independent random variables,

rLlZpnT02ZU

now it's time to wake up.

rLlZpnT02ZU

So we have the variance of something

rLlZpnT02ZU

that looks like 1 over n, the sum from i equal 1 to n of Xi.

rLlZpnT02ZU

So it's of the form variance of a constant times

rLlZpnT02ZU

a random variable.

rLlZpnT02ZU

So the first thing I'm going to do is pull out the constant.

rLlZpnT02ZU

But we know that the variance leaves on the square scale,

rLlZpnT02ZU

so when I pull out a constant outside of the variance,

rLlZpnT02ZU

it comes out with a square.

rLlZpnT02ZU

The variance of a times X is a-squared

rLlZpnT02ZU

times the variance of X, so this is equal to 1

rLlZpnT02ZU

over n squared times the variance of the sum.

rLlZpnT02ZU

So now we want to always do what we want to do.

rLlZpnT02ZU

So we have the variance of the sum.

rLlZpnT02ZU

We would like somehow to say that this

rLlZpnT02ZU

is the sum of the variances.

rLlZpnT02ZU

And in general, we are not allowed to say that,

rLlZpnT02ZU

but we are because my Xi's are actually independent.

rLlZpnT02ZU

So this is actually equal to 1 over n squared sum from i equal

rLlZpnT02ZU

1 to n of the variance of each of the Xi's.

rLlZpnT02ZU

And that's by independence, so this is basic probability.

rLlZpnT02ZU

And now, what is the variance of Xi's where again they're

rLlZpnT02ZU

all the same distribution, so the variance of Xi

rLlZpnT02ZU

is the same as the variance of X1.

rLlZpnT02ZU

And so each of those guys has variance what?

rLlZpnT02ZU

What is the variance of a Bernoulli?

rLlZpnT02ZU

We've said it once.

rLlZpnT02ZU

It's theta times 1 minus theta.

rLlZpnT02ZU

And so now I'm going to have the sum of n times a constant,

rLlZpnT02ZU

so I get n times the constant divided by n squared,

rLlZpnT02ZU

so one of the n's is going to cancel.

rLlZpnT02ZU

And so the whole thing here is actually

rLlZpnT02ZU

equal to theta, 1 minus theta divided by n.

rLlZpnT02ZU

So if I'm interested in the quadratic risk--

rLlZpnT02ZU

and again, I should just say risk,

rLlZpnT02ZU

because this is the only risk we're going