Common Errors in Statistics (and How to Avoid Them), 4th Edition

BAYESIAN METHODOLOGY

Since being communicated to the Royal Society in 1763 by Reverend Thomas Bayes,⁵ the eponymous Theorem has exerted a near-fatal attraction on those exposed to it.⁶ Much as a bell placed on the cat would magically resolve so many of the problems of the average house mouse, Bayes’ straightforward, easily grasped mathematical formula would appear to provide the long-awaited basis for a robotic judge that is free of human prejudice.

On the plus side, Bayes’ Theorem offers three main advantages:

1. Simplifies the combination of a variety of different kinds of evidence, lab tests, animal experiments, and clinical trials, and serves as an effective aid to decision making.

2. Permits evaluating evidence in favor of a null hypothesis. And with very large samples, a null hypothesis is not automatically rejected.

3. Provides dynamic flexibility during the conduct of an experiment; sample sizes can be modified, measuring devices altered, subject populations changed, and end points redefined.

Suppose we have in hand a set of evidence E = {E₁, E₂, … , E_n}, and thus have determined the conditional probability Pr{A | E} that some event A is true. A might be the event that O. J. Simpson killed his ex-wife, that the Captain of the Exxon Valdez behaved recklessly, or some other incident whose truth or falsehood we wish to establish. An additional piece of evidence E_{n +1} now comes to light. Bayes’ Theorem tell us that

where ∼A, read not A, is the event that A did not occur. ...