DEFINITION of Posterior Probability
Posterior probability is the revised probability of an event occurring after taking into consideration new information. Posterior probability is calculated by updating the prior probability by using Bayes' theorem. In statistical terms, the posterior probability is the probability of event A occurring given that event B has occurred.
BREAKING DOWN Posterior Probability
Bayes' theorem can be used in many applications, such as medicine, finance and economics. In finance, Bayes' theorem can be used to update a previous belief once new information is obtained. Prior probability represents what is originally believed before new evidence is introduced, and posterior probability takes this new information into account. As a simple example to envision posterior probability, suppose there are three acres of land with labels A, B and C. One acre has reserves of oil below its surface, while the other two do not. The prior probability of oil in acre C is one-third, or 33%. A drilling test is conducted on acre B, and the results indicate that no oil is present at the location. With acre B eliminated, the posterior probability of acre C containing oil becomes 0.5, or 50%.
Bayes' Theorem Formula
The formula to calculate a posterior probability of A occurring given that B occurred:
P(A | B) = P(B | A)*P(A) / P(B)
1) A and B are events; 2) P(B) is greater than zero; 3) P(B | A) is the probability of B occurring given that A is true and 4) P(A) and P(B) are the probabilities of A occurring and B occurring independently of each other.