**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 $\Phi\left(z\right)$.