In the case of the drunk, I'd like to know if the experimenter was also drinking. I've encountered reports of remarkable happenings with similar evidence, like the hurricane party which deposited a number of sailors in Tampa Bay during a hurricane. Reasoning must have been impaired to choose that time to go sailing.
Prior probability distributions can be abused. I have seen examples where Bayesian methods have been abused to "weight the dice" so that you would need an endless series of experiments to reject an hypothesis the experimenter preferred. Unfortunately, the frequentist alternative can also be abused to set prior probability to zero while claiming objectivity.
The classic medical application of Bayesian methods took place when
Bradford Hill investigated the linkage between smoking and lung cancer in the 1950s. His opponent was smoker, and consultant for tobacco companies, R.A. Fisher. Fisher had been part of the "new synthesis" of Darwinian evolution based on population genetics, and was a formidable force in applied statistics. Fisher delayed official recognition of health dangers of smoking by about 10 years. Sir Austin Bradford Hill turned out to be right, and not by accident.
Bayesian methods were already in use during WWII in methods Turing developed to break German ciphers, using what he called "the weight of evidence". This technique was eventually published in an academic paper by a surviving colleague, but public revelations of the importance of cryptanalysis had to wait 30 years. (Similar methods were used in less-classified work by Abraham Wald on sequential testing.) The critical insight is that probabilities are measures of information, and will change when available information changes. This confounds people indoctrinated with Fisher's approach, as happened with
The Monty Hall Problem.
There are some very sophisticated ways of applying Bayesian methods, as in biochemical cascades where it is hard to detect some chemicals before they are converted to others. (The same thing happens in particle physics, where you have real trouble detecting things like neutrinos.) What I have to impress on people is that the logic involved is completely inflexible. These assumptions should not be under test. If, however, you have an independent basis for knowing that x->y->z, you can use this to deduce that finding y or z means x must have been present earlier. This can save you from using up large numbers of laboratory animals -- or patients.
