HCOL 195 10/26/09

Reminder: The project handout just had some ideas, you aren’t restricted to them and I am delighted when a group works on something completely new.

Note that there’s a difference between Bayes and frequentist ideas. In particular, in Bayesian thought it is perfectly legitimate to talk about the probability of something that just happens to be unknown to us, but is perfectly certain. For example, we can talk about the probability that the Nile is over 1000 miles long…think of it as a bet, for example, what odds would you be willing to give someone else to take either side of a bet that the Nile is over 1000 miles long? If you would be willing to bet at double or nothing, for example, then you think that it’s a 50% probability that the Nile is over 1000 miles long. Frequentists aren’t allowed to use probability this way.

In particular, you should not be thinking of the utilities and losses we’ve been discussing in terms of many, many bets. For example, if you own a house, you shouldn’t be thinking of a lot of identical situations where your house may or may not burn down in a given year. Either it does or it doesn’t. If the house is worth, say, $200,000, and there is a 1 in 1000 chance that it burns down in a year, then the fair value of the expectation of a bet with an insurance company (the premium) that the house will burn down is $200, but you would never get an insurance company to take that bet. They will require significantly more, to cover their fixed expenses and (over many different houses with many different customers) to have a high probability of making a profit for their shareholders.

When we discussed a sure $100,000 versus a 50:50 bet of $1,000,000 or nothing, many preferred the sure thing. This is because the additional $900,000 isn’t (for these folks) as valuable as the first $100,000.

I’ve already posted the link to the podcast on Elinor Ostrom (see previous post). The podcast says everything.

I noted that the patient is the one that has the responsibility to make decisions about medical care. This is because the patient is the one that suffers the consequences. The role of the doctor is to explain the treatments, the consequences, and how likely the various outcomes are, in a way that the patient can understand well enough to make informed decisions. Similarly, lawyers cannot tell their customers what they should do. They are like doctors: Explain the law, and the probable consequences if the client decides on various different courses of action.

There are no riskless actions. Even just lying in bed has risks. Crossing the street, you could get hit by a bus and killed. You are willing to do this for a meal worth a few dollars only because the risk of getting hit by a bus is very low. This can be used in principle to decide on how valuable (in dollars) you think your life is.

One student showed that it is better to buy two lottery tickets on different numbers than to buy two tickets on the same number.

We then discussed the problem of which is worse: Convicting an innocent person (CI) or acquitting a guilty one (AG).

If you acquit a guilty one, then that person will be free to reoffend; on the other hand, the fact that we have his fingerprints, DNA, picture, and other information about him might serve to deter him to some degree and will make him easier to catch.

If you convict an innocent one, then the real culprit is still at large, free to commit another crime. We don’t have any accurate information about the culprit (no DNA, no picture, no name, no prints), and the police will have stopped looking for him. So he may be more prone to commit other crimes. In addition, there is an innocent person in prison, which is another bad thing.

On balance, it seems that CI is worse than AG.

This leads us to consider the following decision tree (assuming that the good outcomes, CG and AI are equally good with a loss of 0). We adopt a loss of 1 for the intermediate decision, AG. We put the worst one, CI, at the top of the chance node. I asked you to think about what value of p would make you indifferent between the two decisions. We’ll discuss it next time.

Tree to decide on how bad CI is relative to AG

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