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Let's now go through
another example,

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which will be a little
more challenging.

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00:00:06,770 --> 00:00:09,040
We're going to revisit
an old problem.

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We have a coin that has
an unknown bias, Theta.

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And we have a prior
distribution on this Theta.

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We fix some positive
integer, n, we flip a coin

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n times, that has
this unknown bias.

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And we record the
number of heads.

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On the basis of the number of
heads that have been observed,

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we wish to estimate the
bias, Theta, of the coin.

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To make things more
concrete, we're

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going to assume a prior
distribution on Theta that

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is uniform on the unit interval.

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Now, this is a problem we
have considered before.

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We have calculated the expected
value of Theta given X.

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And we did find that
the expected value takes

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this particular form.

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Now, notice that this is
a linear function of X.

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And if it turns out the
least mean squares estimator

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is a linear function
of X, then we're

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guaranteed, since
this is the best,

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that this is also the
best within the class

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of linear estimators.

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So we immediately
have the conclusion

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that the linear
least mean squares

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estimator is this
particular function of X.

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So there's not much left to do.

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On the other hand,
just for practice,

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let us derive this answer
directly from the formulas

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that we have for the linear
least mean squares estimator,

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and see whether we're going
to get the same answer.

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So we want to use this formula.

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And in order to apply this
formula, all that we have to do

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is to calculate these expected
values, this variance,

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and this covariance.

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So now let's move on to this
particular calculational

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exercise.

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Let's start by
writing down what we

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know about the random variables
involved in this problem.

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About Theta, we know
that it is uniform.

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And so it has a mean of
1/2 and a variance of 1/12.

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About X, what we know
is the following.

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If you fix the bias of
the coin, then the number

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of heads you're going
to obtain in n flips

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has a binomial distribution,
with parameters n and Theta.

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But of course, Theta itself
is a random variable.

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So for this reason, this is
a conditional distribution.

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But within the
conditional universe,

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we know the mean and the
variance of a binomial,

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and they are as follows.

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The mean of a binomial is n
times the bias of the coin.

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But because we're talking
about the conditional universe,

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this is a conditional
expectation.

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And it's a random
variable, because it's

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affected by the value of
the random variable Theta.

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And similarly, for
the variance, it's

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the usual formula for the
variance of a binomial,

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except that now the bias
itself is a random variable.

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So now let's continue with the
calculation of the quantities

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that we need for the
formula for our estimator.

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Let's start with the
expected value of X.

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Since we know the
conditional expectation of X,

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we can use the law of
iterated expectations.

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The unconditional expectation
is the expected value

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of the conditional expectation,
which is n times Theta.

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And since the mean of Theta
is 1/2, we obtain n/2.

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Let us now continue with the
calculation of the variance.

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There are different ways
that we can calculate it.

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One could be the law
of total variance.

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But we will take the
alternative approach,

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which is to use the general
formula for the variance,

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that the variance is
equal to the expected

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value of the square
of a random variable,

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minus the square of
the expected value.

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We know the expected value of X.

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So all that's left
is to calculate

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the expected value of X squared.

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How are we going calculate it?

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Well, we know the conditional
distribution of X.

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So it should be
easy to calculate

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00:04:13,800 --> 00:04:17,820
the conditional
expectation of X squared

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in the conditional
universe, and then

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use the law of
iterated expectations

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to obtain the
unconditional expectation.

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So now, we need to calculate
this conditional expectation

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here.

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How do we do it?

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The expected value of a
square of a random variable

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is always equal to the variance
of that random variable

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plus the square of
the expected value.

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We're going to
use this property,

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but we're going to use it
in the conditional universe.

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So in the conditional
universe, this

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00:04:50,580 --> 00:04:52,900
is going to be equal
to the variance,

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00:04:52,900 --> 00:04:57,670
in the conditional universe,
which is n times Theta times 1

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00:04:57,670 --> 00:05:04,020
minus Theta, plus the square
of the expected value of X,

97
00:05:04,020 --> 00:05:07,300
but the expected value in the
conditional universe, which

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is this quantity--
n times Theta.

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00:05:10,050 --> 00:05:16,210
So we obtain another term--
n squared, Theta squared.

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So now we can go back to
our previous calculation,

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and right here,
the expected value

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of this expression,
which is n times Theta.

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And then we have some
Theta squared terms.

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One is n squared.

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The other is a minus n.

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00:05:39,159 --> 00:05:46,880
So we obtain plus n squared
minus n Theta squared.

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00:05:51,380 --> 00:05:54,409
The expected value
of n times Theta

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is n times the expected
value of Theta, which is 1/2.

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So we obtain a factor of n/2.

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But then we have this
additional term here.

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We need the expected
value of Theta squared.

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What is it?

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Well, since we know the mean
and the variance of Theta,

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we can calculate the expected
value of Theta squared.

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It is equal to the variance
plus the square of the mean.

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And this evaluates to 1/3.

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So from here, we're
going to obtain

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plus n squared minus
n divided by 3.

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And you can rewrite this
term in a different way

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by collecting first the
n terms, n/2 minus n/3.

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This gives us n/6.

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And then there's
the n squared term,

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which is n squared over 3.

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Now that we found the
expected value of X squared,

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we can go back to
this calculation.

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We have n/6 plus
n squared over 3

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minus the square of the
expected value of X,

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which is this expression here.

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So we obtain minus
n squared over 4.

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1/3 minus 1/4-- that makes 1/12.

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So we obtain n/6 plus
n squared over 12.

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Or another way of writing this
is n times n plus 2 over 12.

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And this completes
the calculation

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of the variance of X.

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The last quantity that's
left for us to calculate

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is the covariance
of Theta with X.

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We're going to calculate it
using the alternative formula

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for the covariance, which is
the expectation of the product

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minus the product
of the expectations.

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We have the expectations,
but we do not

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00:08:19,300 --> 00:08:21,540
have the expectation
of the product.

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So we need to calculate it.

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Once more, it's going
to be the same trick.

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We're going to
condition on Theta,

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and then use the law of
iterated expectations.

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00:08:33,070 --> 00:08:35,350
So the law of
iterated expectations,

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00:08:35,350 --> 00:08:38,789
when you condition on
Theta, takes this form.

148
00:08:38,789 --> 00:08:41,230
And to continue
here, we need to find

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00:08:41,230 --> 00:08:45,220
this conditional
expectation in the inside.

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00:08:45,220 --> 00:08:49,030
This conditional
expectation-- what is it?

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00:08:49,030 --> 00:08:51,630
If I give you Theta,
then you know Theta.

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00:08:51,630 --> 00:08:53,470
It is becoming now a constant.

153
00:08:53,470 --> 00:08:55,410
There's nothing
random to it, so it

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00:08:55,410 --> 00:08:58,600
can be pulled outside
the expectation.

155
00:08:58,600 --> 00:09:03,780
And we obtain Theta times the
conditional expectation of X.

156
00:09:03,780 --> 00:09:06,740
We know what the conditional
expectation of X is.

157
00:09:06,740 --> 00:09:08,070
It's n times Theta.

158
00:09:08,070 --> 00:09:13,330
So from here, we obtain overall,
a term n times Theta squared.

159
00:09:13,330 --> 00:09:15,940
So now we can go back here.

160
00:09:15,940 --> 00:09:17,780
We have the expected value.

161
00:09:17,780 --> 00:09:20,890
And the term in the
inside-- we just found it.

162
00:09:20,890 --> 00:09:23,390
It's n times Theta squared.

163
00:09:23,390 --> 00:09:25,750
And since the expected
value of Theta squared

164
00:09:25,750 --> 00:09:28,695
is 1/3, from here,
we obtain n/3.

165
00:09:32,670 --> 00:09:36,040
And now we can go back,
finally, to the calculation

166
00:09:36,040 --> 00:09:37,330
of the covariance.

167
00:09:37,330 --> 00:09:43,160
It's going to be n/3, minus
the expected value of Theta,

168
00:09:43,160 --> 00:09:48,350
which is 1/2, times the expected
value of X, which is n/2.

169
00:09:48,350 --> 00:09:52,310
So it's minus n over four.

170
00:09:52,310 --> 00:09:54,330
And this evaluates to n/12.

171
00:09:57,880 --> 00:10:01,240
So we have succeeded in
calculating all the quantities

172
00:10:01,240 --> 00:10:04,830
that are needed in the
formula for the linear least

173
00:10:04,830 --> 00:10:06,045
mean squares estimator.

174
00:10:08,710 --> 00:10:12,540
We can now take those values
that we have just found

175
00:10:12,540 --> 00:10:15,620
and substitute them
into this formula.

176
00:10:15,620 --> 00:10:18,650
And after a little bit of
algebra and moving terms

177
00:10:18,650 --> 00:10:22,820
around, everything simplifies
to this expression.

178
00:10:22,820 --> 00:10:25,670
Just to verify that
this makes sense,

179
00:10:25,670 --> 00:10:28,122
what is the
coefficient next to X?

180
00:10:28,122 --> 00:10:30,820
It's the covariance
divided by the variance.

181
00:10:30,820 --> 00:10:34,320
n/12 divided by
this expression--

182
00:10:34,320 --> 00:10:36,050
this n cancels that n.

183
00:10:36,050 --> 00:10:37,410
This 12 cancels that 12.

184
00:10:37,410 --> 00:10:40,460
We're left with an n plus
2 in the denominator.

185
00:10:40,460 --> 00:10:43,520
And indeed, the coefficient
that multiplies X

186
00:10:43,520 --> 00:10:47,960
is the term n plus 2
in the denominator.

187
00:10:47,960 --> 00:10:51,280
And you can similarly verify
that the constant term as well

188
00:10:51,280 --> 00:10:53,870
is the correct one.

189
00:10:53,870 --> 00:10:58,400
So of course, this answer is
what we had found in the past

190
00:10:58,400 --> 00:11:02,160
to be the optimal,
the least mean squares

191
00:11:02,160 --> 00:11:08,310
estimator of X. As we discussed
earlier, when this is linear

192
00:11:08,310 --> 00:11:11,970
in X, it has to be the same as
the linear least mean squares

193
00:11:11,970 --> 00:11:13,050
estimator.

194
00:11:13,050 --> 00:11:15,530
So this answer is
not a surprise,

195
00:11:15,530 --> 00:11:17,930
but it was an interesting
and perhaps useful

196
00:11:17,930 --> 00:11:21,440
exercise to go through the
details of this calculation

197
00:11:21,440 --> 00:11:24,030
to see what it
takes to figure out

198
00:11:24,030 --> 00:11:27,360
the different terms
in this formula.