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 — cs-677sp2010:decisions-lab [2014/12/12 13:40] (current)ryancha created 2014/12/12 13:40 ryancha created 2014/12/12 13:40 ryancha created Line 1: Line 1: + == Discrete EVSI == + '''​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. + + === Discrete Expected Utility === + + 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. + + === Conditional Expected Utilities === + + 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 + + === Expected Value of Sample Information === + + 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). + + == Burglar Alarm == + + 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 [http://​www.d.umn.edu/​~rmaclin/​cs8751/​Notes/​chapter14a.pdf 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$1000. Should I call the police if John calls me? If Mary calls? If both call? + + Now suppose that I can call the alarm company and find out if the alarm is going off for a fee (x dollars). For what values of x (if any) is it worth calling the alarm company if Mary has called? Note that you do not need to find the '''​exact'''​ cut off point, plot the net expected utility for maybe 10 possible x's that show the cut off point. + + '''​Please use your mcmc code to answer this question'''​. You may want to add a little code to automate the collection of expected utilities. Please think carefully about how to use your code to solve this problem. ​ + + == Continuous Expected Utility == + + This is just a continuous expected utility. + + If you find yourself actually trying to solve the integral using your calculus book, you have forgotten to apply a something you know about expectations!  ​ + + '''​Do this by hand'''​ + + Suppose: + + :$\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? + + == Continuous EVSI == + + '''​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 $\Phi\left(z\right)$.