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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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