Posterior probability

Lore & Background

Posterior probability is a type of conditional probability that results from updating the prior probability with information summarized by the likelihood via an application of Bayes' rule. From an epistemological perspective, the posterior probability contains everything there is to know about an uncertain proposition, such as a scientific hypothesis or parameter values, given prior knowledge and a mathematical model describing the observations available at a particular time. After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating.

Reader's Guide

In Bayesian statistics, the posterior probability distribution describes epistemic uncertainty about statistical parameters conditional on observed data. From a given posterior distribution, various point and interval estimates can be derived, such as the maximum a posteriori (MAP) or the highest posterior density interval (HPDI). However, while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated. The calculation involves multiplying the prior probability distribution by the likelihood function, then dividing by the normalizing constant. The example of a school with 60% boys and 40% girls illustrates the concept: given that all boys wear trousers and half of girls do, the posterior probability that a trouser-wearing student is a girl is 25%. This demonstrates how Bayes' theorem updates initial beliefs based on observed evidence.

Did You Know?

• Posterior probability is defined as p(θ|x) = [p(x|θ) / p(x)] × p(θ).
• The normalizing constant p(x) is calculated by integrating or summing p(x|θ)p(θ) over all possible θ.
• In the school example, the posterior probability that a trouser-wearing student is a girl is 25%.
• The posterior distribution is often not tractable and requires analytical or numerical approximation.