ABSTRACT
Bayesian statistics is based on Bayes’ rule, which is named after the Rev. Thomas Bayes, who discussed it in a paper published posthumously in 1763. We can introduce the concept as follows. Let A and B be two events. For example, suppose we are sampling leaves in a wheat –eld to determine whether the mean nitrogen level of the plants in the –eld exceeds the recommended minimum for an adequate supply of nitrogen. Let A be the event that the mean N level of a particular leaf sample exceeds the recommended minimum, and let B be the event that mean N level over all of the plants in the –eld meets the recommended minimum. The joint probability, denoted by P{A, B} is the probability that both events A and B occur. The conditional probability that A occurs given that B occurs is denoted by P{A|B}, and one can similarly de–ne the conditional probability P{B|A} that B occurs given that A occurs. The laws of probability state that (Larsen and Marx, 1986, p. 42)
P A B P B A P A P A B P B{ , } { } { } { } { },= =| | (14.1)
and these can be used to formally de–ne the conditional probability. Bayes’ rule is obtained by dividing the second equation in (14.1) by P{A} (Koop, 2003, p. 1):
P BA
P A B P B P A
{ } { } { }
{ } .|
| = (14.2)
If we have observed the event A, that the mean N level of the sample exceeds the minimum, then Bayes’ rule gives us a means to compute the probability that B occurs, that is, that the –eld mean N level exceeds the minimum.
