Bayes’ Theorem: The concept of conditional probability has been introduced earlier. We noted that the conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. We used P (B│A) to denote the conditional probability of event B occurring, given that event A has already occurred. The following formula was provided for finding P(B│A):
P (A and B)
P(B│A) = ————–
P (A)
The conditional probability of B given A can be found by assuming that event A has occurred and, working under the assumption, calculating the probability that event B will occur.
In this section, we extend the discussion of conditional probability to include applications of Bayes’ theorem (or Bayes’rule), which we use for revising a probability value based on additional information that is later obtained. One key to understanding the essence of Bayes’ theorem is to recognise that we are dealing with sequential events, whereby new additional information is obtained for a subsequent event, and that new information is used to revise the probability of the initial event. In this context, the terms prior probability and posterior probability are commonly used.
Definitions
A prior probability is an initial probability value originally obtained before any additional information is obtained.
A posterior probability is a probability value that has been revised by using additional information that is later obtained.
Bayes’ Theorem
The probability of event A given that event B has subsequently occurred, is
P (A). P(B│A)
P(A│B) = —————————————-
[P(A).P(B│A)] + [P(A̅). P(B│A̅)]
