Chapter 6

Bayes Rule

# 6.1 Transposed Conditionals

This chapter is devoted to what is surely the most interesting rule in probability, with an overall importance that makes it fit to rank alongside the basic equations of Einstein or the fundamental rules of genetics. But first a few examples that are included to demonstrate the need for the rule.

No one who has absorbed the thesis of this book will confuse

The notation makes it apparent that they are different, reversing the orders of the two events, E and F. In the first probability, E is an uncertain event whose belief is being measured supposing, or knowing, that F is true (plus an unstated ). In the second probability, E, far from being uncertain, is supposed or known to be true; it is F that is uncertain and whose belief is being assessed. Despite the obvious differences, people are continually confusing one probability with the other. They are termed transposed conditionals, because E and F have been transposed in the two probabilities, each taking it in turn to be the conditional. They are sometimes referred to as Janus examples after the Roman god with two heads looking in opposite directions. Notice that the confusion occurs in ordinary logic, as in this example from a newspaper, the person's name being changed. “If it is ...