Class, 11/5

We continued our discussion on the General's Dilemma problem. Here, the situation is that there is a desperate battle to be won, and the results will depend on how many soldiers that the general can get to the battle. The Objective (in the PrOACT agenda) is to win the battle. If 600 soldiers get through, then there is a 90% probability that the battle will be won. If 200 make it through, then the probabilitiy is 50%, but if none make it through, then the probability is only 20%.

We assigned a utility of 100 to winning, and 0 to losing. We decided that the expected utility would be maximized if the "200 for sure" choice was made. The decision tree had at the bottom the decision between routes, then if "200 for sure" a probability choice of 0 if lost (80% prob) and 100 if won (20% prob). On the other branch we had to have a 2/3 chance of every soldier being lost versus an 1/3 chance that all would get through. Then (given that fact) whe had to put another probability branch with the appropriate probability of winning the battle (if none get through or if all get through).

When we ran the utilities through the tree backwards to to the left, we found that the general should go with the "200 for sure" route.

We also experimented with the idea that the utilities might be different for the general if he valued lives saved more than lives lost (that is, if the premises of the problem were not just "win the battle" but also "save our soldiers' lives.") That would be done by decreasing the utilities when more soldiers didn't make it through. But fiddling with the numbers didn't seem to make a difference as to the decision. This doesn't mean that some choice of utilities wouldn't change the result, just that it isn't obvious how to do this.

I then opened the discussion to questions.

We discussed priors, especially when the priors depend on the state of nature. In particular, is it necessary to "normalize" the prior (make it add up to 1)? The answer is that in most cases this is not necessary. The reason is that (in our spreadsheet scheme) if you multiply the prior by a constant number then the joint will also be multiplied by the same number, and the sum of the joints (the marginal likelihood) will also be multiplied by the same number, so when you divide the joints by the marginal likelihood, they will have the same constant multiplier, which will cancel out.

The exception is when you are testing a precise hypothesis (a coin is fair prob=1/2, a special die is fair, prob=1/3) against a vague alternative (a coin is not fair, prob has to have its own prior, etc.) In that case, it's important to normalize the priors and careful attention is required to do it right.

We briefly discussed the medical example, when we have tests on an old and new drug. The whole idea here is to compute the probability that the new drug is better than the old one. We've been using an approximation that sets the cure rate of a drug to particular values, 0.05, 0.15, 0.25, ..., 0.95. We know that if we make the division finer, we'll get a better approximation, but here we're trying to learn the principles. So, if we test the old drug (A) and get a certain set of posterior probabilities on the cure rate of the drug, and test the new drug (B) and get a different set of posterior probabilities, then we can set up a table of the joint posterior probabilities of the cure rates of each drug by simply multiplying the probs of the cure rates of each drug. We can do this because the two experiments were done on different and randomly selected individual patients, so the probs are independent. Then, it's just a matter of identifying which slots belong to (cure rate of B is greater than cure rate of A), adding them up (we add when we have (cure rate of A=0.05 AND cure rate of B=0.15) OR (cure rate of A=0.05 AND cure rate of B=0.25) OR ....) This gives us the posterior probability that drug B is better than drug A.

I think this is all we discussed, but if I have left something out important and you want me to address it before the test, post now and I'll respond.