Editing Posterior probability (section)

{{short description|Conditional probability used in Bayesian statistics}}{{Bayesian statistics}}

The '''posterior probability''' is a type of [[conditional probability]] that results from [[Bayesian updating|updating]] the [[prior probability]] with information summarized by the [[likelihood function|likelihood]] via an application of [[Bayes' rule]].<ref>{{cite book |first=Ben |last=Lambert |chapter=The posterior – the goal of Bayesian inference |pages=121–140 |title=A Student's Guide to Bayesian Statistics |location= |publisher=Sage |year=2018 |isbn=978-1-4739-1636-4 }}</ref> From an [[Bayesian epistemology|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.<ref>{{cite thesis |last=Grossman |first=Jason |title=Inferences from observations to simple statistical hypotheses |type=PhD thesis |publisher=University of Sydney |year=2005 |hdl=2123/9107 }}</ref> After the arrival of new information, the current posterior probability may serve as the prior in another round of Bayesian updating.<ref>{{Cite web |last=Etz |first=Alex |date=2015-07-25 |title=Understanding Bayes: Updating priors via the likelihood |url=https://alexanderetz.com/2015/07/25/understanding-bayes-updating-priors-via-the-likelihood/ |access-date=2022-08-18 |website=The Etz-Files |language=en}}</ref>

In the context of [[Bayesian statistics]], the '''posterior [[probability distribution]]''' usually describes the epistemic uncertainty about [[statistical parameter]]s conditional on a collection of observed data. From a given posterior distribution, various [[point estimate|point]] and [[interval estimate]]s can be derived, such as the [[Maximum a posteriori estimation|maximum a posteriori]] (MAP) or the [[highest posterior density interval]] (HPDI).<ref>{{cite book |first=Jeff |last=Gill |title=Bayesian Methods: A Social and Behavioral Sciences Approach |edition=Third |publisher=Chapman & Hall |year=2014 |isbn=978-1-4398-6248-3 |chapter=Summarizing Posterior Distributions with Intervals |pages=42–48 }}</ref> But while conceptually simple, the posterior distribution is generally not tractable and therefore needs to be either analytically or numerically approximated.<ref>{{cite book |first=S. James |last=Press |chapter=Approximations, Numerical Methods, and Computer Programs |pages=69–102 |title=Bayesian Statistics : Principles, Models, and Applications |location=New York |publisher=John Wiley & Sons |year=1989 |isbn=0-471-63729-7 }}</ref>