# Bayes theorem

Imagine that there are 20 patients in a waiting room.  A proportion of 10/20 has diabetes, a proportion of 12/20 is obese and a proportion of 9/20 is both obese and diabetic.  It is a matter of simple arithmetic that:

10/20 x 9/10 = 9/20 = 12/20 x 9/12

This can also be written as probabilities:

0.5 x 0.9 = 0.45 = 0.6 x 0.75

The above equations can be ‘rearranged’ by dividing each side of the equations by 10/20 or 0.5.  This gives Bayes theorem:

9/10 = 12/20 x 9/12 ÷ 10/20

or

0.9 = 0.6 x 0.75 ÷ 0.5

For patients in the room the proportion of 10/20 = 0.5 is called the ‘prior probability’ of having diabetes and 12/20 = 0.6 is the ‘prior probability’ of being obese.  The proportion of 9/12 = 0.75 is called the ‘likeliness’ of being obese when a patient is known to have diabetes.

Bayes theorem tells us that the probability of diabetes when someone in the room is known to be obese is 9/10 or 0.9:

9/10 = 12/20 x 9/12 ÷ 10/20

or

0.9 = 0.6 x 0.75 ÷ 0.5

Bayesian statisticians use this relationship to calculate the probability of replicating a study result if the study was repeated using an infinite number of patients by taking the ‘prior probability’ of replicating the result into account.