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rLlZpnT02ZU
|
to be actually looking at.
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rLlZpnT02ZU
|
Yeah.
|
rLlZpnT02ZU
|
This parenthesis should
really stop here.
|
rLlZpnT02ZU
|
I really wanted to put
quadratic in parenthesis.
|
rLlZpnT02ZU
|
So the risk of this guy is what?
|
rLlZpnT02ZU
|
Well, it's the expectation of
x bar n minus theta squared.
|
rLlZpnT02ZU
|
And we know it's the
square of the variance,
|
rLlZpnT02ZU
|
so it's the square
of the bias, which
|
rLlZpnT02ZU
|
we know is 0, so it's 0 squared
plus the variance, which
|
rLlZpnT02ZU
|
is theta, 1 plus theta--
|
rLlZpnT02ZU
|
1 minus theta divided by n.
|
rLlZpnT02ZU
|
So it's just theta, 1
minus theta divided by n.
|
rLlZpnT02ZU
|
So this is just summarizing the
performance of an estimator,
|
rLlZpnT02ZU
|
which is the random variable.
|
rLlZpnT02ZU
|
I mean, it's complicated.
|
rLlZpnT02ZU
|
If I really wanted
to describe it,
|
rLlZpnT02ZU
|
I would just tell you
the entire distribution
|
rLlZpnT02ZU
|
of this random variable.
|
rLlZpnT02ZU
|
But now what I'm doing
is I'm saying, well,
|
rLlZpnT02ZU
|
let's just take this random
variable, remove theta from it,
|
rLlZpnT02ZU
|
and see how small the
fluctuations around theta--
|
rLlZpnT02ZU
|
the squared fluctuations around
theta are in expectation.
|
rLlZpnT02ZU
|
So that's what the
quadratic risk is doing.
|
rLlZpnT02ZU
|
And in a way, this
decomposition,
|
rLlZpnT02ZU
|
as the sum of the bias
square and the variance,
|
rLlZpnT02ZU
|
is really telling you that--
|
rLlZpnT02ZU
|
it is really accounting for
the bias, which is, well,
|
rLlZpnT02ZU
|
even if I had an infinite
amount of observations,
|
rLlZpnT02ZU
|
is this thing doing
the right thing?
|
rLlZpnT02ZU
|
And the other thing is
actually the variance,
|
rLlZpnT02ZU
|
so for finite number
of observations,
|
rLlZpnT02ZU
|
what are the fluctuations?
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
Then you can see that those
things, bias and variance,
|
rLlZpnT02ZU
|
are actually very different.
|
rLlZpnT02ZU
|
So I don't have any
colors here, so you're
|
rLlZpnT02ZU
|
going to have to really
follow the speed--
|
rLlZpnT02ZU
|
the order in which
I draw those curves.
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
So let's find--
|
rLlZpnT02ZU
|
I'm going to give you three
candidate estimators, so--
|
rLlZpnT02ZU
|
estimators for theta.
|
rLlZpnT02ZU
|
So the first one is
definitely Xn bar.
|
rLlZpnT02ZU
|
That will be a good
candidate estimator.
|
rLlZpnT02ZU
|
The second one is going to
be 0.5, because after all,
|
rLlZpnT02ZU
|
why should I bother if
it's actually going to be--
|
rLlZpnT02ZU
|
right?
|
rLlZpnT02ZU
|
So for example, if
I ask you to predict
|
rLlZpnT02ZU
|
the score of some
candidate in some election,
|
rLlZpnT02ZU
|
then since you know it's
going to be very close to 0.5,
|
rLlZpnT02ZU
|
you might as well just throw
0.5 and you're not going
|
rLlZpnT02ZU
|
to be very far from reality.
|
rLlZpnT02ZU
|
And it's actually going
to cost you 0 time and $0
|
rLlZpnT02ZU
|
to come up with that.
|
rLlZpnT02ZU
|
So sometimes maybe
just a good old guess
|
rLlZpnT02ZU
|
is actually doing
the job for you.
|
rLlZpnT02ZU
|
Of course, for
presidential elections
|
rLlZpnT02ZU
|
or something like this,
it's not very helpful
|
rLlZpnT02ZU
|
if your prediction
is telling you this.
|
rLlZpnT02ZU
|
But if it was
something different,
|
rLlZpnT02ZU
|
that would be a good way to
generate some close to 1/2.
|
rLlZpnT02ZU
|
For a coin, for example,
if I give you a coin,
|
rLlZpnT02ZU
|
you never know.
|
rLlZpnT02ZU
|
Maybe it's slightly biased.
|
rLlZpnT02ZU
|
But the good guess, just
looking at it, inspecting it,
|
rLlZpnT02ZU
|
maybe there's something
crazy happening
|
rLlZpnT02ZU
|
with the structure
of it, you're going
|
rLlZpnT02ZU
|
to guess that it's 0.5 without
trying to collect information.
|
rLlZpnT02ZU
|
And let's find another one,
which is, well, you know,
|
rLlZpnT02ZU
|
I have a lot of observations.
|
rLlZpnT02ZU
|
But I'm recording couples
kissing, but I'm on a budget.
|
rLlZpnT02ZU
|
I don't have time to
travel all around the world
|
rLlZpnT02ZU
|
and collect some people.
|
rLlZpnT02ZU
|
So really, I'm just going
to look at the first couple
|
rLlZpnT02ZU
|
and go home.
|
rLlZpnT02ZU
|
So my other estimator
is just going to be X1.
|
rLlZpnT02ZU
|
I just take the first
observation, 0, 1,
|
rLlZpnT02ZU
|
and that's it.
|
rLlZpnT02ZU
|
So now I'm going--
|
rLlZpnT02ZU
|
I want to actually understand
what the behavior of those guys
|
rLlZpnT02ZU
|
is.
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
So we know-- and so we know
that for this guy, the bias is 0
|
rLlZpnT02ZU
|
and the variance
is equal to theta,
|
rLlZpnT02ZU
|
1 minus theta divided by n.
|
rLlZpnT02ZU
|
What is the bias
of this guy, 0.5?
|
rLlZpnT02ZU
|
AUDIENCE: 0.5.
|
rLlZpnT02ZU
|
AUDIENCE: 0.5 minus theta?
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: 0.5
minus theta, right.
|
rLlZpnT02ZU
|
So the bias, 0.5 minus theta.
|
rLlZpnT02ZU
|
What is the variance
of this guy?
|
rLlZpnT02ZU
|
What is the variance of 0.5?
|
rLlZpnT02ZU
|
AUDIENCE: It's 0.
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: 0.
|
rLlZpnT02ZU
|
Right.
|
rLlZpnT02ZU
|
It's just a
deterministic number,
|
rLlZpnT02ZU
|
so there's no
fluctuations for this guy.
|
rLlZpnT02ZU
|
What is the bias?
|
rLlZpnT02ZU
|
Well, X1 is actually--
|
rLlZpnT02ZU
|
just for simplicity,
I can think of it
|
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