Class, 10/22

OK, so today I revisited the insurance problem, but from the point of view of expected loss rather than expected utility. It's not any different, but I drew a picture on the blackboard that showed that there was a range in insurance premiums between the minimum premium that the insuarnce company would sell the insurance policy for (p=m/h, see previous posting for definitions) and the maximum amount the homeowner would buy it for (p=loss(m)/loss(h)). The homeowner hopes that between these two limits, there will be competition between insurance companies that will give him or her a good deal on insurance.

We then discussed testing various hypotheses based on data observed.

First, we discussed testing a coin which may be fair or unfair.

We decided that if it was fair, a reasonable prior would be P(fair)=0.5, and P(unfair)=0.5.

But then, what does P(unfair) mean? If the coin is fair, it is supposed to be Heads with probability 0.5. But if it is unfair? What is the probability? We decided to split the probability up equally amongst the possibilities, which we chose to be 0.05, 0.15, 0.25,...0.85, 0.95 with each possibility having prior probability 0.05 (after some discussion that reminded us that the the total probability has to be 1, and already we have expended 0.5 on the "null hypothesis" that the coin is fair.)

So we set up a "spreadsheet" calculation. We discussed how to actually do it if we were doing it with Excel.

We didn't actually do the calculation, but I will tell you that the result is: With 60 heads and 40 tails, the probability of obtaining a result that extreme or more extreme (60 or more heads, or 40 or less tails) is about 0.05, but the probability that the coin is fair, given that we have observed the data (60 heads and 40 tails) is about 0.5.

This is very interesting. The standard statistical test of statistical significance, how extreme the result is, is very different from what the Bayesian result is.

We also discussed the problem of estimating the probability that an unknown proportion (here, the bias of the coin, or in the problem set, the cure rate of the new drug) is greater than some fixed value (say 0.2). The spreadsheet is the same except for no special picking out of 0.5; this means we crossed out this line and used 0.1 for the alternatives 0.5, 0.15,...0.95, and to determine the probability that the the new drug is better than the old one, we just add the probabilities for the states of nature that are greater than 0.2. (Again we didn't do the actual calculation.

We finished with a challenge: How to decide, if you are on a jury, whether to convict or acaquit a defendant in a criminal case. More clearly, what does "beyond a reasonable doubt" mean?