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rLlZpnT02ZU
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because the theta that you
have over there is really-- so
|
rLlZpnT02ZU
|
in the definition of
the risk, the theta
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rLlZpnT02ZU
|
that you have here
if you're unbiased
|
rLlZpnT02ZU
|
is really the
expectation of theta hat.
|
rLlZpnT02ZU
|
So that's really
just the variance.
|
rLlZpnT02ZU
|
So the risk is
really telling you
|
rLlZpnT02ZU
|
how much fluctuations I
have around my expectation
|
rLlZpnT02ZU
|
if unbiased.
|
rLlZpnT02ZU
|
But actually here, it's telling
you how much fluctuations
|
rLlZpnT02ZU
|
I have in average around theta.
|
rLlZpnT02ZU
|
So if you understand the
notion of variance as being--
|
rLlZpnT02ZU
|
AUDIENCE: [INAUDIBLE]
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rLlZpnT02ZU
|
PHILIPPE RIGOLLET: What?
|
rLlZpnT02ZU
|
AUDIENCE: Like
variance on average.
|
rLlZpnT02ZU
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PHILIPPE RIGOLLET: No.
|
rLlZpnT02ZU
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AUDIENCE: No.
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: It's
just like variance.
|
rLlZpnT02ZU
|
AUDIENCE: Oh, OK.
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: So when you--
|
rLlZpnT02ZU
|
I mean, if you claim you
understand what variance is,
|
rLlZpnT02ZU
|
it's telling you
what is the expected
|
rLlZpnT02ZU
|
squared fluctuation
around the expectation
|
rLlZpnT02ZU
|
of my random variable.
|
rLlZpnT02ZU
|
It's just telling you on
average how far I'm going to be.
|
rLlZpnT02ZU
|
And you take the square because
you want to cancel the signs.
|
rLlZpnT02ZU
|
Otherwise, you're
going to get 0.
|
rLlZpnT02ZU
|
AUDIENCE: Oh, OK.
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: And
here it's saying, well,
|
rLlZpnT02ZU
|
I really don't care what the
expectation of theta hat is.
|
rLlZpnT02ZU
|
What I want to get
to is theta, so I'm
|
rLlZpnT02ZU
|
looking at the expectation
of the squared fluctuations
|
rLlZpnT02ZU
|
around theta itself.
|
rLlZpnT02ZU
|
If I'm unbiased, it
coincides with the variance.
|
rLlZpnT02ZU
|
But if I'm biased, then I
have to account for the fact
|
rLlZpnT02ZU
|
that I'm really
not computing the--
|
rLlZpnT02ZU
|
AUDIENCE: OK.
|
rLlZpnT02ZU
|
OK.
|
rLlZpnT02ZU
|
Thanks.
|
rLlZpnT02ZU
|
PHILIPPE RIGOLLET: OK?
|
rLlZpnT02ZU
|
All right.
|
rLlZpnT02ZU
|
Are there any questions?
|
rLlZpnT02ZU
|
So here, what I really
want to illustrate
|
rLlZpnT02ZU
|
is that the risk
itself is a function
|
rLlZpnT02ZU
|
of theta most of the times.
|
rLlZpnT02ZU
|
And so for different
thetas, some estimators
|
rLlZpnT02ZU
|
are going to be
better than others.
|
rLlZpnT02ZU
|
But there's also
the entire range
|
rLlZpnT02ZU
|
of estimators, those
that are really biased,
|
rLlZpnT02ZU
|
but the bias can
completely vanish.
|
rLlZpnT02ZU
|
And so here, you see
you have no bias,
|
rLlZpnT02ZU
|
but the variance can be large.
|
rLlZpnT02ZU
|
Or you have 0 bias--
|
rLlZpnT02ZU
|
you have a bias, but
the variance is 0.
|
rLlZpnT02ZU
|
So you can actually
have this trade-off
|
rLlZpnT02ZU
|
and you can find things that are
in the entire range in general.
|
rLlZpnT02ZU
|
So those things are
actually-- those trade-offs
|
rLlZpnT02ZU
|
between bias and variance are
usually much better illustrated
|
rLlZpnT02ZU
|
if we're talking about
multivariate parameters.
|
rLlZpnT02ZU
|
If I actually look
at a parameter which
|
rLlZpnT02ZU
|
is the mean of some multivariate
Gaussian, so an entire vector,
|
rLlZpnT02ZU
|
then the bias is going to--
|
rLlZpnT02ZU
|
I can make the bias
bigger by, for example,
|
rLlZpnT02ZU
|
forcing all the coordinates of
my estimator to be the same.
|
rLlZpnT02ZU
|
So here, I'm going
to get some bias,
|
rLlZpnT02ZU
|
but the variance
is actually going
|
rLlZpnT02ZU
|
to be much better, because
I get to average all
|
rLlZpnT02ZU
|
the coordinates for this guy.
|
rLlZpnT02ZU
|
And so really, the
bias/variance trade-off
|
rLlZpnT02ZU
|
is when you have multiple
parameters to estimate,
|
rLlZpnT02ZU
|
so you have a vector
of parameters,
|
rLlZpnT02ZU
|
a multivariate
parameter, the bias
|
rLlZpnT02ZU
|
increases when you're trying
to pull more information
|
rLlZpnT02ZU
|
across the different
components to actually have
|
rLlZpnT02ZU
|
a lower variance.
|
rLlZpnT02ZU
|
So the more you average,
the lower the variance.
|
rLlZpnT02ZU
|
That's exactly what
we've illustrated.
|
rLlZpnT02ZU
|
As n increases, the
variance decreases,
|
rLlZpnT02ZU
|
like 1 over n or theta,
1 minus theta over n.
|
rLlZpnT02ZU
|
And so this is how it
happens in general.
|
rLlZpnT02ZU
|
In this class, it's mostly
one-dimensional parameter
|
rLlZpnT02ZU
|
estimation, so it's going to be
a little harder to illustrate
|
rLlZpnT02ZU
|
that.
|
rLlZpnT02ZU
|
But if you do, for example,
non-parametric estimation,
|
rLlZpnT02ZU
|
that's all you do.
|
rLlZpnT02ZU
|
There's just bias/variance
trade-offs all the time.
|
rLlZpnT02ZU
|
And in between, when you have
high-dimensional parametric
|
rLlZpnT02ZU
|
estimation, that
happens a lot as well.
|
rLlZpnT02ZU
|
OK.
|
rLlZpnT02ZU
|
So I'm just going to go quickly
through those two remaining
|
rLlZpnT02ZU
|
slides, because we've
actually seen them.
|
rLlZpnT02ZU
|
But I just wanted you to have
somewhere a formal definition
|
rLlZpnT02ZU
|
of what a confidence
interval is.
|
rLlZpnT02ZU
|
And so we fixed a statistical
model for n observations, X1
|
rLlZpnT02ZU
|
to Xn.
|
rLlZpnT02ZU
|
The parameter theta
here is one-dimensional.
|
rLlZpnT02ZU
|
Theta is a subset
of the real line,
|
rLlZpnT02ZU
|
and that's why I
talk about intervals.
|
rLlZpnT02ZU
|
An interval is a
subset of the line.
|
rLlZpnT02ZU
|
If I had a subset
of R2, for example,
|
rLlZpnT02ZU
|
that would no longer be called
an interval, but a region,
|
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