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rLlZpnT02ZU | that I get different
samples every time |
rLlZpnT02ZU | should somewhat vanish. |
rLlZpnT02ZU | And so what I want is to have a
small bias, hopefully a 0 bias. |
rLlZpnT02ZU | If this thing is 0, then we see
that the estimator is unbiased. |
rLlZpnT02ZU | So this is definitely
a property that we |
rLlZpnT02ZU | are going to be looking
for in an estimator, |
rLlZpnT02ZU | trying to find them
to be unbiased. |
rLlZpnT02ZU | But we'll see that it's
actually maybe not enough. |
rLlZpnT02ZU | So unbiasedness should
not be something |
rLlZpnT02ZU | you lose your sleep over. |
rLlZpnT02ZU | Something that's slightly
better is the risk, really |
rLlZpnT02ZU | the quadratics risk,
which is expectation of-- |
rLlZpnT02ZU | so if I have an
estimator, theta hat, |
rLlZpnT02ZU | I'm going to look at the
expectation of theta hat n |
rLlZpnT02ZU | minus theta squared. |
rLlZpnT02ZU | And what we showed last time
is that we can actually-- |
rLlZpnT02ZU | by inserting in there,
adding and removing |
rLlZpnT02ZU | the expectation of
theta hat, we actually |
rLlZpnT02ZU | get something where
this thing can |
rLlZpnT02ZU | be decomposed as the square
of the bias plus the variance, |
rLlZpnT02ZU | which is just the expectation of
theta hat minus its expectation |
rLlZpnT02ZU | squared. |
rLlZpnT02ZU | That came from
the fact that when |
rLlZpnT02ZU | I added and removed the
expectation of theta hat |
rLlZpnT02ZU | in there, the
cross-terms cancel. |
rLlZpnT02ZU | All right. |
rLlZpnT02ZU | So that was the bias squared,
and this is the variance. |
rLlZpnT02ZU | And so for example, if the
quadratic risk goes to 0, |
rLlZpnT02ZU | then that means that
theta hat converges |
rLlZpnT02ZU | to theta in the L2 sense. |
rLlZpnT02ZU | And here we know that if
we want this to go to 0, |
rLlZpnT02ZU | since it's the sum of
two positive terms, |
rLlZpnT02ZU | we need to have both
the bias that goes to 0 |
rLlZpnT02ZU | and the variance
that goes to 0, so we |
rLlZpnT02ZU | need to control both
of those things. |
rLlZpnT02ZU | And so there is usually
an inherent trade-off |
rLlZpnT02ZU | between getting a small bias
and getting a small variance. |
rLlZpnT02ZU | If you reduce one too much, then
the variance of the other one |
rLlZpnT02ZU | is going to-- |
rLlZpnT02ZU | then the other one is going
to increase, or the opposite. |
rLlZpnT02ZU | That happens a lot, but not so
much, actually, in this class. |
rLlZpnT02ZU | So let's just look at
a couple of examples. |
rLlZpnT02ZU | So am I planning-- |
rLlZpnT02ZU | yeah. |
rLlZpnT02ZU | So examples. |
rLlZpnT02ZU | So if I do, for example, X1,
Xn, there are iid Bernoulli. |
rLlZpnT02ZU | And I'm going to
write it theta so |
rLlZpnT02ZU | that we keep the same notation. |
rLlZpnT02ZU | Then theta hat, what
is the theta hat |
rLlZpnT02ZU | that we proposed many times? |
rLlZpnT02ZU | It's just X bar, Xn bar,
the average of Xi's. |
rLlZpnT02ZU | So what is the bias of this guy? |
rLlZpnT02ZU | Well, to know the bias, I
just have to remove theta |
rLlZpnT02ZU | from the expectation. |
rLlZpnT02ZU | What is the
expectation of Xn bar? |
rLlZpnT02ZU | Well, by linearity
of the expectation, |
rLlZpnT02ZU | it's just the average
of the expectations. |
rLlZpnT02ZU | But since all my Xi's are
Bernouilli with the same theta, |
rLlZpnT02ZU | then each of this guy is
actually equal to theta. |
rLlZpnT02ZU | So this thing is actually
theta, which means |
rLlZpnT02ZU | that this isn't biased, right? |
rLlZpnT02ZU | Now, what is the
variance of this guy? |
rLlZpnT02ZU | So if you forgot the
properties of the variance |
rLlZpnT02ZU | for sum of independent
random variables, |
rLlZpnT02ZU | now it's time to wake up. |
rLlZpnT02ZU | So we have the
variance of something |
rLlZpnT02ZU | that looks like 1 over n, the
sum from i equal 1 to n of Xi. |
rLlZpnT02ZU | So it's of the form
variance of a constant times |
rLlZpnT02ZU | a random variable. |
rLlZpnT02ZU | So the first thing I'm going
to do is pull out the constant. |
rLlZpnT02ZU | But we know that the variance
leaves on the square scale, |
rLlZpnT02ZU | so when I pull out a constant
outside of the variance, |
rLlZpnT02ZU | it comes out with a square. |
rLlZpnT02ZU | The variance of a
times X is a-squared |
rLlZpnT02ZU | times the variance of
X, so this is equal to 1 |
rLlZpnT02ZU | over n squared times
the variance of the sum. |
rLlZpnT02ZU | So now we want to always
do what we want to do. |
rLlZpnT02ZU | So we have the
variance of the sum. |
rLlZpnT02ZU | We would like somehow
to say that this |
rLlZpnT02ZU | is the sum of the variances. |
rLlZpnT02ZU | And in general, we are
not allowed to say that, |
rLlZpnT02ZU | but we are because my Xi's
are actually independent. |
rLlZpnT02ZU | So this is actually equal to 1
over n squared sum from i equal |
rLlZpnT02ZU | 1 to n of the variance
of each of the Xi's. |
rLlZpnT02ZU | And that's by independence,
so this is basic probability. |
rLlZpnT02ZU | And now, what is the variance
of Xi's where again they're |
rLlZpnT02ZU | all the same distribution,
so the variance of Xi |
rLlZpnT02ZU | is the same as the
variance of X1. |
rLlZpnT02ZU | And so each of those
guys has variance what? |
rLlZpnT02ZU | What is the variance
of a Bernoulli? |
rLlZpnT02ZU | We've said it once. |
rLlZpnT02ZU | It's theta times 1 minus theta. |
rLlZpnT02ZU | And so now I'm going to have
the sum of n times a constant, |
rLlZpnT02ZU | so I get n times the constant
divided by n squared, |
rLlZpnT02ZU | so one of the n's
is going to cancel. |
rLlZpnT02ZU | And so the whole
thing here is actually |
rLlZpnT02ZU | equal to theta, 1 minus
theta divided by n. |
rLlZpnT02ZU | So if I'm interested
in the quadratic risk-- |
rLlZpnT02ZU | and again, I should
just say risk, |
rLlZpnT02ZU | because this is the
only risk we're going |
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