Do this one by hand, but you can check it with your code if you really want to. This is like the oil example from class.

let $P\left(x\right) = 0.2$ for a Boolean Random Variable X.

Assume that you have your choice between two utilities, both functions of X:

$U\left(1,x\right)= 400$ and $U(1,\neg x)= 2$

$U\left(2,x\right)= 20$ and $U(2,\neg x)= 100$

Compute the Expected Utilities and state what choice you would make.

Suppose that X influences another Random Variable, Y, in the following way:

$p\left(Y=1|x\right) = 0.2$

$p\left(Y=2|x\right) = 0.4$

$p(Y=1|\neg x) = 0.6$

$p(Y=2|\neg x) = 0.3$

Note that Y can take three values 1,2, or 3

Compute the conditional probabilities:

Note that we will be computing many probabilities in this and subsequent sections. You may compute the joint of x and y and then sum the needed Values out of your joint or use Bayes' law directly.

• $p\left(x|Y=1\right)$
• $p\left(x|Y=2\right)$
• $p\left(x|Y=3\right)$

Use these probabilities to compute the conditional expected utilities:

• $E\left(U\left(1,x\right)|Y=1\right)$
• $E\left(U\left(2,x\right)|Y=1\right)$
• $E\left(U\left(1,x\right)|Y=2\right)$
• $E\left(U\left(2,x\right)|Y=2\right)$
• $E\left(U\left(1,x\right)|Y=3\right)$
• $E\left(U\left(2,x\right)|Y=3\right)$

What choice would you make in each of the following cases and what utility would you expect in each case.

• Y=1
• Y=2
• Y=3

Compute the following probabilities:

• $p\left(Y=1\right)$
• $p\left(Y=2\right)$
• $p\left(Y=3\right)$

What is the Expected Posterior Utility? What is the Expected Value of Sample Information?

This sequence of questions led you through the computations needed for EVSI. Please note that if I were to put an EVSI question on the test, I would not lead you through the computation step by step as I have done here. You will need to understand the process on your own. You may also get a slightly more complex net (like with 3 nodes).

The Burglar Alarm example is described on pages 493 and 494 in Russell and Norvig. If you don't have the book, look at slides 5 and 6 on this pdf.

Suppose that the city of Berkley charges $25 for the police to respond to a false alarm and that the deductible on the insurance for burglaries is $1000. Should I call the police if John calls me? If Mary calls? If both call?

Now suppose that I can call the alarm company and find out if the alarm is going off for a fee (x dollars). For what values of x (if any) is it worth calling the alarm company if Mary has called? Note that you do not need to find the exact cut off point, plot the net expected utility for maybe 10 possible x's that show the cut off point.

Please use your mcmc code to answer this question. You may want to add a little code to automate the collection of expected utilities. Please think carefully about how to use your code to solve this problem.

This is just a continuous expected utility.

If you find yourself actually trying to solve the integral using your calculus book, you have forgotten to apply a something you know about expectations!

Do this by hand

Suppose:

:$\theta \sim N\left(1,2\right)$

:$U\left(1,\theta\right)=\frac{1}{4}\cdot \theta + 4$

:$U\left(2,\theta\right)=-2\cdot \theta + 4$

What decision would you make?

Do this by hand (see the Continuous EVSI page for help)

Using the model from class for normally distributed $\theta \sim N\left(1,2\right)$ (as above) and $y \sim N\left(\theta,2\right)$ with linear cost functions as given above, compute the EVSI.

You may use a stats package or a www applet (or a book!!) to compute $\phi\left(z\right)$ and $\Phi\left(z\right)$.