Probably Not, 2nd Edition

11Bayesian Statistics

Bayes Theorem

In Chapter 10, we derived and used the following formula for conditional probability:

(11.1)

Rewriting Eq. (11.1) as

(11.2)

Since P(D and T) must be equal to P(T and D),

(11.3)

from which

(11.4)

Equation (11.4) is known as Bayes theorem, after the mathematician who first described it, more than 300 years ago. As we shall soon see, it leads to an entirely different definition of probability and approach to many problems than we have seen thus far.

There is an interesting history of debate between followers of the two approaches. In this book, we shall present both approaches and give examples of their utility and not weigh in on either side of the debate.

However, rather than beginning with formal definitions, we'll start by using Bayes formula in a few example problems. Along the way, we'll rename the terms in Bayes formula to align with this new way of thinking.

Repeating the medical test problem of Chapter 10: there is a deadly disease that affects .1% of the population. A test for the disease yields 2% false‐positive and 2% false‐negative results. ...