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rLlZpnT02ZU | to be actually looking at. |
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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