What is a Posterior Probability?

Posterior probability measures the likelihood that an event will occur given that a related event has already occurred. It is a modification of the original probability or the probability without further information, which is called prior probability (先验概率，普通意义下的概率). Posterior probability is calculated using Bayes’ Theorem. Financial modeling of stock portfolios is a common application of posterior probability in finance. It is sometimes difficult to accurately assign probabilities to events, limiting posterior probability’s usefulness.

In order to calculate posterior probability, the conditional probability of two dependent events can be examined. Let A be the target event, then P(A) is the a priori probability. Let B be a second event that is dependent, or is related to the event A, with probability P(B) （A 先发生，B 后发生，B依赖于A）. Furthermore, let the likelihood of event B occurring, given that A occurs, be P(B|A).

Using Bayes’ Theorem, the posterior probability P(A|B) can be calculated. The theory states: P(A|B)=P(B|A)∗P(A)P(B)P(A|B)=P(B|A)∗P(A)P(B)P(A|B) = \frac{P(B|A)*P(A)}{ P(B)}. Note that if events A and B are independent, then their joint probability is P(A|B)=P(A)P(A|B)=P(A)P(A|B) = P(A). This means that their posterior and prior probabilities are identical, since the event B has no effect on the event A.

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