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.
In short, he could easily be a frequentist and use the same method and come to the same conclusion.
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.
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!
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 explic...
Why is Bayes' Rule useful? Most explanations of Bayes explain the how of Bayes: they take a well-posed mathematical problem and convert given numbers to desired numbers. While Bayes is useful for calculating hard-to-estimate numbers from easy-to-estimate numbers, the quantitative use of Bayes requires the qualitative use of Bayes, which is noticing that such a problem exists. When you have a hard-to-estimate number that you could figure out from easy-to-estimate numbers, then you want to use Bayes. This mental process of testing beliefs and searching for easy experiments is the heart of practical Bayesian thinking. As an example, let us examine 1 Kings 3:16-28:
Notice that Solomon explicitly identified competing hypotheses, raising them to the level of conscious attention. When each hypothesis has a personal advocate, this is easy, but it is no less important when considering other uncertainties. Often, a problem looks clearer when you branch an uncertain variable on its possible values, even if it is as simple as saying "This is either true or not true."
Solomon considers the empirical consequences of the competing hypotheses, searching for a test which will favor one hypothesis over another. When considering one hypothesis alone, it is easy to find tests which are likely if that hypothesis is true. The true mother is likely to say the child is hers; the true mother is likely to be passionate about the issue. But that's not enough; we need to also estimate how likely those results are if the hypothesis is false. The false mother is equally likely to say the child is hers, and could generate equal passion. We need a test whose results significantly depend on which hypothesis is actually true.
Witnesses or DNA tests would be more likely to support the true mother than the false mother, but they aren't available. Solomon realizes that the claimant's motivations are different, and thus putting the child in danger may cause the true mother and false mother to act differently. The test works, generates a large likelihood ratio, and now his posterior firmly favors the first claimant as the true mother.