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Posterior (p₍θ┃X₎) ○◂｜Definition｜1st｜20251119205401-00-⌔

Posterior probability - Wikipedia

Posterior probability

The 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.1 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.2 After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating.3

In the context of Bayesian statistics, the posterior probability distribution usually describes the epistemic uncertainty about statistical parameters conditional on a collection of 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).4 But while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated.5

Printed 2026-06-28.

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Footnotes

  1. Lambert, Ben (2018). “The posterior – the goal of Bayesian inference”. A Student’s Guide to Bayesian Statistics. Sage. pp. 121–140. ISBN 978-1-4739-1636-4. ↩

  2. Grossman, Jason (2005). Inferences from observations to simple statistical hypotheses (PhD thesis). University of Sydney. hdl:2123/9107. ↩

  3. Etz, Alex (2015-07-25). “Understanding Bayes: Updating priors via the likelihood”. The Etz-Files. Retrieved 2022-08-18. ↩

  4. Gill, Jeff (2014). “Summarizing Posterior Distributions with Intervals”. Bayesian Methods: A Social and Behavioral Sciences Approach (Third ed.). Chapman & Hall. pp. 42–48. ISBN 978-1-4398-6248-3. ↩

  5. Press, S. James (1989). “Approximations, Numerical Methods, and Computer Programs”. Bayesian Statistics: Principles, Models, and Applications. New York: John Wiley & Sons. pp. 69–102. ISBN 0-471-63729-7. ↩

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