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Posterior probability

The posterior probability is one of the quantities involved in Bayes' rule. It is the conditional probability of a given event, computed after observing a second event whose conditional and unconditional probabilities were known in advance. It is computed by revising the prior probability, that is, the probability assigned to the first event before observing the second event.

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Definition

The following is a more formal definition.

Definition Let A and $B$ be two events whose prior probabilities [eq1] and [eq2] are known. Assume that also the conditional probability [eq3] is known. By Bayes' rule, we have that[eq4]The conditional probability [eq5] thus computed is called posterior probability.

In other words, the posterior probability is the conditional probability [eq6] assigned to the event A after receiving the information that the event $B$ has happened, and it is computed by exploiting the knowledge of the conditional probability [eq7] and of the prior probabilities [eq8] and [eq2].

Example

Suppose that an individual is extracted at random from a population of men. The probability of extracting a married individual is 50%. The probability of extracting a childless individual is 40%. The conditional probability that an individual is childless given that he is married is equal to 20%. If the individual we extract at random from the population turns out to be childless, what is the conditional probability that he is married? This conditional probability is called posterior probability and it can be computed by using Bayes' rule above.

The quantities involved in the computation are[eq10]

The posterior probability is[eq11]

More details

A more detailed explanation of the concept of posterior probability can be found in the lecture entitled Bayes' rule.

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