Inferring a Parameter of Uniform Part 2

Inferring a Parameter of Uniform Part 2


Welcome back. So now we’re going to finish
the rest of this problem. For part e, we’ve calculated
what the map and LMS estimators are. And now we’re going to calculate
what the conditional mean squared error is. So it’s a way to measure how
good these estimators are. So let’s start out
generically. For any estimator theta hat,
the conditional MSE is– conditional mean
squared error– is equal to this. It’s the estimator minus the
actual value squared conditioned on X being equal
to some little x. So the mean squared error. So you take the error, which is
the difference between your estimator and the true
value, square it, and then take the mean. And it’s conditioned on the
actual value of what x is. Or, conditioned on the
data that you get is. So to calculate this, we use
our standard definition of what conditional expectation
would be. So it’s theta hat minus
theta squared. And we weight that by the
appropriate conditional PDF, which in this case would
be the posterior. And we integrate this from x– from theta equals x
to theta equals 1. Now, we can go through some
algebra and this will tell us that this is theta hat squared
minus 2 theta hat theta minus plus theta squared. And this posterior we know from
before is 1 over theta times absolute value
of log x d theta. And when we do out this
integral, it’s going to be– we can split up in to three
different terms. So there’s theta hat squared
times this and you integrate it. But in fact, this is just
a conditional density. When you integrate it from x to
1, this will just integrate up to 1 because it is
a valid density. So the first term is just
theta hat squared. Now, the second term is you can
pull out of 2 theta hat and integrate theta times 1
over theta times absolute value of log of x d
theta from x to 1. And then the last one is
integral of theta squared 1 over theta absolute value of
log x d theta from x to 1. OK, so we can do some more– with some more calculus, we get
a final answer is this. So this will integrate to
1 minus x over absolute value of log x. And this will integrate to 1
minus x squared over 2 times absolute value of log x. So this tells us for any generic
estimate theta hat, this would be what the
conditional mean squared error would be. Now, let’s calculate what it
actually is for the specific estimates that we actually
came up with. So for the MAP rule, the
estimate of theta hat is just equal to x. So when we plug that into
this, we get that the conditional MSE is just equal to
x squared minus 2x 1 minus x absolute value of log x plus 1
minus x squared over 2 times absolute value of log of x. And for the LMS estimate,
remember this was equal to– theta hat was 1 minus x over
absolute value of log x. And so when you plug this
particular theta hat into this formula, what you get is that
the conditional mean squared error is equal to 1 minus x
squared over 2 times absolute value of log of x minus
1 minus x over log of x quantity squared. So these two expressions tells
us what the mean squared error is for the MAP rule
and the LMS rule. And it’s kind of hard to
actually interpret exactly which one is better based on
just these expressions. So it’s helpful to plot out
what the conditional mean squared error is. So we’re plotting for x. For each possible actual
data that we observe– data point that we observe,
what is the mean squared error? So let’s do the MAP
rule first. The MAP rule would look
something like this. And it turns out that the LMS
rule is better, and it will look like this dotted line
here on the bottom. And so it turns out that if your
metric for how good your estimate is is the conditional
mean squared error, then LMS is better than MAP. And this is true because LMS
is actually designed to actually minimize what this
mean squared error is. And so in this case, the LMS
estimator should have a better mean squared error than
the map estimator. OK, now the last part of the
problem, we calculate one more type of estimator, which is
the linear LMS estimator. So notice that the LMS estimator
was this one. It was 1 minus x over absolute
value of log of x. And this is not linear in x,
which means sometimes it’s difficult to calculate. And so what we do is we tried to
come up with a linear form of this, something that is like
ax plus b, where a and b are some constant numbers. But that also does well in terms
of having a small mean squared error. And so we know from the class
that in order to calculate the linear LMS, the linear LMS we
know we just need to calculate a few different parts. So it’s equal to the expectation
of the parameter plus the covariance of theta and
x over the variance of x times x minus expectation
of x. Now, in order to do this,
we just need to calculate four things. We need the expectation of
theta, the covariance, the variance, and the expectation
of x. OK, so let’s calculate what
these things are. Expectation of theta. We know that theta is uniformly
distributed between 0 and 1. And so the expectation
of theta is the easiest one to calculate. It’s just 1/2. What about the expectation
of x? Well, expectation of x is a
little bit more complicated. But remember, like in previous
problems, it’s helpful when you have a hierarchy of
randomness to try to use the law of iterated expectations. So the delay, which
is x, is random. But it’s randomness depends
on the actual distribution, which is theta. Which itself is random. And so let’s try to condition
on theta and see if that helps us. OK, so if we knew what theta
was, then what is the expectation of x? Well, we know that given
theta, x is uniformly distributed between
0 and theta. And so the mean would be
just theta over 2. And so this would just be
expectation of theta over 2. And we know this is just 1/2
times the expectation of theta, which is 1/2. So this is just 1/4. Now, let’s calculate
the variance of x. The variance of x takes some
more work because we need to use the law of total variance,
which is this. That the variance of theta is
equal to the expectation of the conditional variance plus
the variance of the conditional expectation. Let’s see if we can figure
out what these different parts are. What is the conditional variance
of x given theta? Well, given theta, x we know
is uniformly distributed between 0 and theta. And remember for uniform
distribution of width c, the variance of that uniform
distribution is just c squared over 12. And so in this case, what is
the width of this uniform distribution? Well, it’s uniformly distributed
between 0 and theta, so the width is theta. So this variance should be
theta squared over 12. OK, what about the expectation
of x given theta? Well, we already argued earlier
that the expectation of x given theta is
just theta over 2. So now let’s fill in the rest. What’s the expectation of
theta squared over 12? Well, that takes a little bit
more work because this is just– you can think
of it as 1/12. You could pull the 1/12
out times the expectation of theta squared. Well, the expectation of theta
squared we can calculate from the variance of theta plus
the expectation of theta quantity squared. Because that is just the
definition of variance. Variance is equal to expectation
of theta squared minus expectation of theta
quantity squared. So we’ve just reversed
the formula. Now, the second half is the
variance of theta over 2. Well, remember when you pull
out a constant from a variance, you have
to square it. So this is just equal to 1/4
times the variance of theta. Well, what is the variance
of theta? The variance of theta is
the variance of uniform between 0 and 1. So the width is 1. So you get 1 squared over 12. And the variance is 1/12. What is the mean of theta? It’s 1/2 when you square
that, you get 1/4. Finally for here, the
variance of theta like we said, is 1/12. So you get 1/12. And now, when you combine all
these, you get that the variance ends up being 7/144. Now we have almost everything. The last thing we need
to calculate is this covariance term. What is the covariance
of theta and x? Well, the covariance we know is
just the expectation of the product of theta and x minus
the product of the expectations. So the expectation of x times
the expectation of theta. All right, so we already know
what expectation of theta is. That’s 1/2. And expectation of x was 1/4. So the only thing that we don’t
know is expectation of the product of the two. So once again, let’s try to
use iterated expectations. So let’s calculate this as
the expectation of this conditional expectation. So we, again, condition
on theta. And minus the expectation
of theta is 1/2. Times 1/4, which is the
expectation of x. Now, what is this conditional
expectation? Well, the expectation
of theta– if you know what theta is, then
the expectation of theta is just theta. You already know what it is, so
you know for sure that the expectation is just
equal to theta. And what is the expectation
of x given theta? Well, the expectation of x given
theta we already said was theta over 2. So what you get is this entire
expression is just going to be equal to theta times theta
over 2, or expectation of theta squared over
2 minus 1/8. Now, what is the expectation
of theta squared over 2? Well, we know that– we already calculated out
what expectation of theta squared is. So we know that expectation
of theta squared is 1/12 plus 1/4. So what we get is we need a 1/2
times 1/12 plus 1/4, which is 1/3 minus 1/8. So the answer is 1/6 minus
1/8, which is 1/24. Now, let’s actually plug
this in and figure out what this value is. So when you get everything– when you combine everything,
you get that the LMS estimator is– the linear LMS estimator
is going to be– expectation of theta is 1/2. The covariance is 1/24. The variance is 7/144. And when you divide that, it’s
equal to 6/7 times x minus 1/4 because expectation
of x is 1/4. And you can simplify this a
little bit and get that this is equal to 6/7 times
x plus 2/7. So now we have three different
types of estimators. The map estimator,
which is this. Notice that it’s kind
of complicated. You have x squared terms. You have more x squared terms. And you have absolute
value of log of x. And then you have the LMS, which
is, again, nonlinear. And now you have something
that looks very simple– much simpler. It’s just 6/7 x plus 2/7. And that is the linear
LMS estimator. And it turns out that you can,
again, plot these to see what this one looks like. So here is our original plot
of x and theta hat. So the map estimator– sorry, the map estimator was
just theta hat equals x. This was the mean squared error
of the map estimator. So the map estimator is just
this diagonal straight line. The LMS estimator looked
like this. And it turns out that the linear
LMS estimator will look something like this. So it is fairly close to
the LMS estimator, but not quite the same. And note, especially that
depending on what x is, if x is fairly close to the 1, you
might actually get an estimate of theta that’s greater
than 1. So for example, if you observe
that Julian is actually an hour late, then x is 1 and your
estimate of theta from the linear LMS estimator
would be 8/7, which is greater than 1. That doesn’t quite make sense
because we know that theta is bounded to be only
between 0 and 1. So you shouldn’t get an estimate
of theta that’s greater than 1. And that’s one of the side
effects of having the linear LMS estimator. So that sometimes you will have
an estimator that doesn’t quite make sense. But what you get instead when
sacrificing that is you get a simple form of the estimator
that’s linear. And now, let’s actually
consider what the performance is. And it turns out that the
performance in terms of the conditional mean squared error
is actually fairly close to the LMS estimator. So it looks like this. Pretty close, pretty close,
until you get close to 1. In which case, it does worse. And it does worse precisely
because it will come up with estimates of theta which
are greater than 1, which are too large. But otherwise, it does pretty
well with a estimator that is much simpler in form than
the LMS estimator. So in this problem, which had
several parts, we actually went through, basically, all
the different concepts and tools within Chapter Eight
for Bayesian inference. We talked about the prior, the
posterior, calculating the posterior using the
Bayes’ rule. We calculated the
MAP estimator. We calculated the
LMS estimator. From those, we calculated what
the mean squared error for each one of those and
compared the two. And then, we looked at the
linear LMS estimator as another example and calculated
what that estimator is, along with the mean squared error
for that and compared all three of these. So I hope that was a good review
problem for Chapter Eight, and we’ll see
you next time.

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