Male and Female Majors at Yale

Dan Gelerntner at Yale has a weblog post listing the most male and the most female majors at Yale. It's interesting that Music makes the male list, and history of science and cognitive science make the female list.

What to Do when Theory and Data Are in Conflict. Realclimate makes a good point:

It is salutary to keep in mind that in many past cases where data conflicted with robust modeling results, it turned out to be the models that were right and the data that was wrong. This was the case for the early satellite reconstructions of twentieth century lower tropospheric temperature, which showed a spurious cooling. It was also the case for early reconstructions of tropical climate during the Last Glacial Maximum, which failed to show the cooling we now know to prevail in that region during glacial times.

Men and Women. My wife told me about a comparison she had heard. Women are like spaghetti on a plate; men are like waffles. A woman's thoughts are in touch with lots of things at once, intricately coiled, and you never know where the end will be. A man's thoughts are compartmentalized. Thus, a man is bothered when a woman starts off by talking about the plumbing and ends up talking about her niece's wedding; he wants to separate things out and resolve one problem at a time.

Part-Whole Bias--The Embedding Effect. ``Contingent valuation'' is a survey technique used in cost-benefit analysis for public goods. These goods are not traded in the market, so market prices cannot be used to value them. Instead, the analyst asks a sample of people how much they are willing to pay for the public good, which is often some environmental good such as wildlife preservation. Surveys are notoriously bad at eliciting true valuations, and the contingent valuation method has been much criticized. Diamond \& Hausman (1994) survey some of these criticisms, including that of ``part-whole bias'', or ``the embedding effect''. The clearest example is from a study by Desvouges, Johnson, Dunford, Hudson, Wilson \& Boyle (1993) which asked some people how much they would pay to stop the killing of 2,000 birds, some people 20,000, and some people 200, 000. The answers were all roughly the same, even though presumably it is worth spending more to save 200,000 birds than to save 2,000. Diamond and Hausman suggest that respondents were not really saying how much they valued birds, but were giving themselves a good feeling by donating, even if only in the abstract, a sum towards wildlife preservation (the ``warm glow effect'' of Andreoni (1989)). People do not view saving 200,000 birds as the addition of one-hundred 2000-bird projects. Similarly, Kemp \& Maxwell (1993) asked one group of people how much they would pay to reduce the risk of oil spills off the coast of Alaska, and found an average valuation of \$85. They asked a different group how much they would pay for a broad range of government programs, and then asked that group to divide and subdivide their willingness to pay for the various items in the package. By the time they broke it down to reducing the risk of oil spills off the coast of Alaska, the value was down to \$0.29. Asking about the oil spills separately gives it a much higher value; the whole is worth more than the sum of the parts. (For a a wide variety of other contingent valuation studies see Frederick \& Fischhoff (1998)). Surveys have their own special problems, but the embedding effect can arise in real decisionmaking too, as Bateman, Munro, Rhodes, Starmer \& Sugden (1997) found in experiments in which subjects traded restaurant vouchers. This should not be surprising. Many people devote considerable effort to budgeting their spending, and such effort would be unnecessary if we were endowed with enough brainpower to costlessly link every consumption decision to every other actual and potential one. Naturally, if we are reminded of other items we could buy-- or told of options that are entirely new to us-- that affects our decisions.

Making Big Choices under Uncertainty. I had a good talk about Christianity today during my visit to Warwick University. We talked about two fallacies of delayed decision. One is that of Buridan's Ass, who, halfway between two equally good mangers of hay, died of starvation because there was no way to choose between them. Lesson: To not choose is a choice in itself, and sometimes worse than not choosing the best alternative. The second fallacy is that of my webpost a few days ago: of choosing X because it is very uncertain which is better, X or Y. This seems silly until I bring in the application: choosing not to pray to God because it is very uncertain which is better, praying to God or (because he might not exist) not praying. It is quite possible to make your choice and pray heartily to a God you are not sure exists, just as you can write letters to someone who might never receive them or spend thousands of dollars on a medical treatment that might have no chance of
working. You may not be able to fix the degree of your belief, and without a strong belief you may find the discipline of following it hard, but you can make the decision.

Logic and Rhetoric. David Hume writes in The Treatise that:

There is an inconvenience which attends all abstruse reasoning, that it may silence, without convincing an antagonist, and requires the same intense study to make us sensible of its force, that was at first requisite for its invention. When we leave our closet, and engage in the common affairs of life, its conclusions seem to vanish, like the phantoms of the night on the appearance of the morning; and ’tis difficult for us to retain even that conviction, which we had attain’d with difficulty.

Bayesian vs. Frequentist Statistical Theory: George and Susan

Susan either likes George or dislikes him. His prior belief is that there is a 50% chance that she likes him. He also believes that if she does, there is an 80% chance she will smile at him, and if she does not, there is a 60% chance. She smiles at him. What should he think of that?

The Frequentist approach says that George should choose the answer which has the greatest likelihood given the data, and so he should believe that she likes him.Click here to read more

It warns, however, that if he follows this plan, and she really doesn't like him, then he will come to the wrong conclusion with 60% probability. Thus, he can't confidently reject the null hypothesis that she dislikes him. (Though he also could not confidently reject the null hypothesis that she likes him-- that plan would lead him to mistakenly rejecting the null 80% of the time!)

The Bayesian answer is that he should do this computation:

Prob(likes|smiles) = Prob (smiles|likes)Prob(likes)/Prob(smiles)

Prob(likes|smiles)= .8*.5/(.8*.5+.6*.5) = .4/(.4+.3) = 4/7.

If he must choose a binary action-- say, to invite her out or not-- then if the losses from each kind of mistake are symmetric, he should act as if she likes him, knowing that with probability 3/7 he will be making a mistake.

Now let's think about where prior information enters. It enters via the Prior, of course-- the 50% chance that she likes him. But it also enters via the Likelihood Function-- the 80% and 60% figures. If we ask how sure George is of his 50% prior, it is in the Likelihood Function that we need to look for the answer. Suppose we want to keep George's belief at 50% but make him very sure of 50%, as if he'd had lots of evidence on each side rather than not knowing Susan at all and guessing based on his experience with other women. How does that show up here?

It shows up as this one smiling incident having very little effect on his belief. If his prior is solid, then he thinks that the data conveys little information. Maybe she would smile with probability 80% if she likes him and 79% if she dislikes him. Moreover, the next time he meets her, her second smile or frown will convey about the same, small, amount of information.

If his prior were loose, the first incident would have a big impact and the second would have a noticeably smaller impact. Maybe she would smile with probability 80% if she likes him and 20% if she dislikes him. Moreover, the next time he meets her, her second smile or frown will convey less information-- maybe the numbers would shift to 80% and 40%.

But my last two paragraphs do not satisfy me. I must go on to other things, so I will leave this post now.