Bayes' Theorem Explained, No Math Required

I was asked by a medical student to explain Bayes' Theorem.  This blog is about lack of common sense in medicine, so it follows that education about first principles will contribute to uncommon sense, and I will oblige.

Bayes' theorem is simply a long or holistic way of looking at the world, one which is more in keeping with reality than the competing frequentist approach.  A Bayesian (a person who subscribes to the logic of Bayes' Theorem) looks at the totality of the data, whereas a frequentist is concerned with just a specific slice of the data such as a test or a discrete dataset.  Frequentist hypothesis testing is where we get P-values from.  Frequentists are concerned with just the data from the current study.  Bayesians are concerned with the totality of the data, and they do meta-analyses, combining data from as many sources as they can.  (But alas they are still reluctant frequentists, because they insist on combining only frequentist datasets, and shun attempts to incorporate more amorphous data such as "what is the likelihood of something like this based on common sense?")

Consider a trial of orange juice (OJ) for the treatment of sepsis.  Suppose that 300 patients are enrolled and orange juice reduces sepsis mortality from 50% to 20% with P<0.001.  The frequentist says "if the null hypothesis is true and there is no effect of orange juice in sepsis, the probability of finding a difference as great or greater than what was found between orange juice and placebo is less than 0.001; thus we reject the null hypothesis."  The frequentist, on the basis of this trial, believes that orange juice is a thaumaturgical cure for sepsis.  But the frequentist is wrong.