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rLlZpnT02ZU
|
as being X1 bar, the
average of itself,
|
rLlZpnT02ZU
|
so that wherever I saw an n for
this guy, I can replace it by 1
|
rLlZpnT02ZU
|
and that will give
me my formula.
|
rLlZpnT02ZU
|
So the bias is
still going to be 0.
|
rLlZpnT02ZU
|
And the variance is going to be
equal to theta, 1 minus theta.
|
rLlZpnT02ZU
|
So now I have those
three estimators.
|
rLlZpnT02ZU
|
Well, if I compare
X1 and Xn bar, then
|
rLlZpnT02ZU
|
clearly I have 0
bias in both cases.
|
rLlZpnT02ZU
|
That's good.
|
rLlZpnT02ZU
|
And I have the variance that's
actually n times smaller when I
|
rLlZpnT02ZU
|
use my n observations
than when I don't.
|
rLlZpnT02ZU
|
So those two guys,
on these two fronts,
|
rLlZpnT02ZU
|
you can actually look
at the two numbers
|
rLlZpnT02ZU
|
and say, well, the first
number is the same.
|
rLlZpnT02ZU
|
The second number is
better for the other guy,
|
rLlZpnT02ZU
|
so I will definitely go for
this guy compared to this guy.
|
rLlZpnT02ZU
|
So this guy is gone.
|
rLlZpnT02ZU
|
But not this guy.
|
rLlZpnT02ZU
|
Well, if I look at the
bias, the variance is 0.
|
rLlZpnT02ZU
|
It's always beating the
variance of this guy.
|
rLlZpnT02ZU
|
And if I look at the bias, it's
actually really not that bad.
|
rLlZpnT02ZU
|
It's 0.5 minus theta.
|
rLlZpnT02ZU
|
In particular, if theta
is 0.5, then this guy
|
rLlZpnT02ZU
|
is strictly better.
|
rLlZpnT02ZU
|
And so you can actually
now look at what
|
rLlZpnT02ZU
|
the quadratic risk looks like.
|
rLlZpnT02ZU
|
So here, what I'm
going to do is I'm
|
rLlZpnT02ZU
|
going to take my
true theta-- so it's
|
rLlZpnT02ZU
|
going to range between 0 and 1.
|
rLlZpnT02ZU
|
And we know that those two
things are functions of theta,
|
rLlZpnT02ZU
|
so I can only understand
them if I plot them
|
rLlZpnT02ZU
|
as functions of theta.
|
rLlZpnT02ZU
|
And so now I'm going
to actually plot--
|
rLlZpnT02ZU
|
the y-axis is going
to be the risk.
|
rLlZpnT02ZU
|
So what is the risk of
the estimator of 0.5?
|
rLlZpnT02ZU
|
This one is easy.
|
rLlZpnT02ZU
|
Well, it's 0 plus the
square of 0.5 minus theta.
|
rLlZpnT02ZU
|
So we know that at theta,
it's actually going to be 0.
|
rLlZpnT02ZU
|
And then it's going
to be a square.
|
rLlZpnT02ZU
|
So at 0, it's going to be 0.25.
|
rLlZpnT02ZU
|
And at 1, it's going
to be 0.25 as well.
|
rLlZpnT02ZU
|
So it looks like this.
|
rLlZpnT02ZU
|
Well, actually, sorry.
|
rLlZpnT02ZU
|
Let me put the 0.5
where it should be.
|
rLlZpnT02ZU
|
OK.
|
rLlZpnT02ZU
|
So this here is the risk of 0.5.
|
rLlZpnT02ZU
|
And we'll write it like this.
|
rLlZpnT02ZU
|
So when theta is very close
to 0.5, I'm very happy.
|
rLlZpnT02ZU
|
When theta gets farther,
it's a little bit annoying.
|
rLlZpnT02ZU
|
And then here, I want to
plot the risk of this guy.
|
rLlZpnT02ZU
|
So now the thing with
the risk of this guy
|
rLlZpnT02ZU
|
is that it will depend on n.
|
rLlZpnT02ZU
|
So I will just pick some
n that I'm happy with just
|
rLlZpnT02ZU
|
so that I can
actually draw a curve.
|
rLlZpnT02ZU
|
Otherwise, I'm going to have to
plot one curve per value of n.
|
rLlZpnT02ZU
|
So let's just say, for
example, that n is equal to 10.
|
rLlZpnT02ZU
|
And so now I need to plot
the function theta, 1 minus
|
rLlZpnT02ZU
|
theta divided by 10.
|
rLlZpnT02ZU
|
We know that theta,
1 minus theta
|
rLlZpnT02ZU
|
is a curve that goes like this.
|
rLlZpnT02ZU
|
It takes value at 1/2.
|
rLlZpnT02ZU
|
It thinks value 1/4.
|
rLlZpnT02ZU
|
That's the maximum.
|
rLlZpnT02ZU
|
And then it's 0 at the end.
|
rLlZpnT02ZU
|
So really, if n is
equal to 1, this
|
rLlZpnT02ZU
|
is what the variance looks like.
|
rLlZpnT02ZU
|
The bias doesn't
count in the risk.
|
rLlZpnT02ZU
|
Yeah.
|
rLlZpnT02ZU
|
AUDIENCE: [INAUDIBLE]
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: Sure.
|
rLlZpnT02ZU
|
Can you move?
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
Are you guys good?
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
So now I have this picture.
|
rLlZpnT02ZU
|
And I know I'm going up to 25.
|
rLlZpnT02ZU
|
And there's a place
where those curves cross.
|
rLlZpnT02ZU
|
So if you're sure--
|
rLlZpnT02ZU
|
let's say you're talking
about presidential election,
|
rLlZpnT02ZU
|
you know that those things
are going to be really close.
|
rLlZpnT02ZU
|
Maybe you're actually
better by predicting 0.5
|
rLlZpnT02ZU
|
if you know it's not
going to go too far.
|
rLlZpnT02ZU
|
But that's for one observation,
so that's the risk of X1.
|
rLlZpnT02ZU
|
But if I look at the
risk of Xn, all I'm doing
|
rLlZpnT02ZU
|
is just crushing
this curve down to 0.
|
rLlZpnT02ZU
|
So as n increases, it's going
to look more and more like this.
|
rLlZpnT02ZU
|
It's the same
curve divided by n.
|
rLlZpnT02ZU
|
And so now I can just
start to understand
|
rLlZpnT02ZU
|
that for different
values of thetas,
|
rLlZpnT02ZU
|
now I'm going to have to be very
close to theta is equal to 1/2
|
rLlZpnT02ZU
|
if I want to start saying
that Xn bar is worse
|
rLlZpnT02ZU
|
than the naive estimator 0.5.
|
rLlZpnT02ZU
|
Yeah.
|
rLlZpnT02ZU
|
AUDIENCE: Sorry.
|
rLlZpnT02ZU
|
I know you explained a little
bit before, but can you just--
|
rLlZpnT02ZU
|
what is an intuitive
definition of risk?
|
rLlZpnT02ZU
|
What is it actually describing?
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET:
So either you can--
|
rLlZpnT02ZU
|
well, when you have an unbiased
estimator, it's simple.
|
rLlZpnT02ZU
|
It's just telling you
it's the variance,
|
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