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novalis comments on Why Bayes? A Wise Ruling - Less Wrong
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Solomon wasn't actually using Bayes here.
The prior here (A has stolen B's baby) is actually quite low. It just doesn't happen very often. Of course, Solomon actually has to consider some extra evidence (B has accused A of stealing her baby). Solomon (by your account) doesn't consider these things at all.
Solomon's analysis only considered the likelihood given a single test.
Irrelevant, because it is certain that one of them attempted to steal the other's baby: the question is whether it was by a midnight baby-swap, or by bearing false witness. What's your prior for the likelihood of attempting by each method conditioned on that an attempt was made? (Note that it could even be a conjunction- when the baby-swap fails, rush to the court and claim that she attempted a baby-swap!)
It could also be an error; maybe B was so blinded by grief that she refused to believe that her own baby had died. (But again, not really the point; the point is that the article has nothing whatsoever to do with Bayes).
I'm wrapping that into 'false witness,' since intentions aren't particularly important about the truth of events.
Would you care to expand on this point? The story obviously predates Bayes, and so doesn't use any of the terminology or explicitly show the process, but it seems to me like a good example of when and how Bayesian thinking would be useful, and if I'm missing something it would probably be rather useful to know.
Bayes goes like this: P(H|E) = P(E|H)*P(H)/P(E). Here, Solomon considers P(E|H) (and P(~E|H)) -- but he doesn't consider P(H) at all. In short, he could easily be a frequentist and use the same method and come to the same conclusion.
I interpreted it as:
P(First Woman)=.5; P(Second Woman)=1-P(First Woman)=.5.
(This is a simplifying assumption, since those aren't actually exhaustive and mutually exclusive.)
He then decides to test Reaction, since he expects that P(Reaction|First Woman) and P(Reaction|~First Woman) are significantly different. The test works, and then he calculates P(First Woman|Reaction) easily.
I don't see the algebra of Bayes as particularly important. Most people shouldn't trust themselves to do algebra correctly without a calculator when important things are on the line, and many practical applications require Bayes nets that are large enough that it is wise to seek computer assistance in navigating them.
To the extent that there is a difference between Bayesians and Frequentists, it's a disagreement about interpretations, not math. It's not like Frequentists disagree with P(H|E) = P(E|H)*P(H)/P(E), or have sworn not to use it!
Part of what I want to do with this post (and any other stories that people can find) is to highlight the qualitative side of Bayes. Someone who understands the algebra but doesn't notice when their life presents them with opportunities to use it is not getting as much out of Bayes as they could.
What I would call the three main components of Bayes are explicitly considering hypotheses, explicitly searching for tests with high likelihood ratios, rather than just high likelihoods, and explicitly incorporating prior information. I'm content with examples that show off only one of those components.
There are at least two meanings to the Bayesian/frequentist debate; one is a disagreement about methods (or at least a different set of tools), and the other is a disagreement about the deeper meaning of probability. This is an article about methods, not meaning. The major difference is that Bayesian methods make the prior explicit. The p-value is, perhaps, the quintessential frequentist statistic. Here, we can easily imagine Solomon publishing his paper in the Ancient Journal of Statistical Law and citing a p-value < 0.001 -- but without knowing the actual P(defendant), we don't know how many times he made the correct decision (in terms of the facts; as noted in another thread, from a child-welfare perspective, the decision was probably correct regardless).
Frequentists also care about high likelihood ratios.
I know I'm nitpicking, but is the prior really that low in Solomon's case ? In our modern times, things like that almost never happen, but Solomon was living in Old Testament times (metaphorically speaking, seeing as the Solomon we're talking about here is just a character in a book). And the Old testament makes few legal distinctions between children and other kinds of property. Stealing them would still be a big deal, but hardly improbable.
OK, then maybe the prior is high. So what? The point is that Solomon didn't consider it. I'm not saying his test was useless or his decision was wrong. I'm saying that the word "Bayes" is being used as an applause light rather than for its meaning!
Yeah, that's why I said I was merely nitpicking.