Other meanings of Posterior probability
Bayesian inference
Posterior probability is the probability assigned to a hypothesis after relevant evidence has been incorporated. It combines an initial or prior probability with the likelihood of observing the evidence under competing hypotheses, producing an updated measure of uncertainty.
Posterior probability is the probability of a hypothesis after incorporating relevant evidence. In Bayesian notation, it is written p(H|E), read as “the probability of H given E,” where H is a hypothesis and E is observed evidence. Bayes’ theorem expresses the update as p(H|E) = p(E|H)p(H) / p(E). The prior probability, p(H), represents uncertainty before the new evidence; the likelihood, p(E|H), measures how compatible the evidence is with the hypothesis; and p(E) is the overall probability of the evidence across the relevant possibilities.1
The denominator makes the resulting probabilities sum to one when several hypotheses are considered. For comparing hypotheses, analysts often use the proportional form p(H|E) ∝ p(E|H)p(H), then normalize the results. Posterior probability is therefore not simply the likelihood of the evidence: it also reflects the starting plausibility of the hypothesis.
Evidence changes posterior probability according to both its diagnostic strength and the prior distribution. Suppose a disease has a prior prevalence of 1%, a test detects the disease in 90% of affected people, and it produces a positive result in 5% of unaffected people. For a randomly selected person with a positive result, the posterior probability is approximately 15.4%, not 90%, because false positives arise from the much larger unaffected population.2
This calculation illustrates the base-rate effect: a test’s sensitivity is not the same quantity as the probability that a positive result indicates disease. More generally, evidence strongly raises a posterior when it is much more likely under the hypothesis than under its alternatives. A likelihood ratio summarizes that contrast. Independent pieces of evidence can often be incorporated sequentially by using one posterior as the next prior, although dependence, selection effects, and shared measurement errors must be modeled rather than ignored.
A posterior probability is conditional on the model and prior assumptions used to obtain it. Priors may represent established frequencies, previous studies, expert judgment, symmetry, or deliberately weak initial information. They can be discrete probabilities over competing hypotheses or continuous distributions over unknown parameters such as a mean, rate, or regression coefficient.3
The likelihood describes how the data-generating process would behave if a proposed parameter or hypothesis were true; it is not, by itself, a probability distribution over hypotheses. A complete Bayesian analysis therefore specifies the sampling model, prior, and observed data. With enough informative data, reasonable priors may produce similar posteriors, but weak data can leave conclusions sensitive to prior choice. Sensitivity analysis—repeating the calculation under plausible alternative priors—is a standard way to reveal that dependence rather than conceal it.
Posterior probability supports decisions under uncertainty by quantifying how plausible hypotheses are after the available evidence is considered. A posterior probability of 0.8 for a hypothesis means that, within the specified model, data, and prior assumptions, the hypothesis has probability 0.8 conditional on the evidence; it does not mean that 80% of the evidence has been explained or that the hypothesis is permanently established.
Applications include medical diagnosis, reliability engineering, genetic analysis, ecological estimation, forecasting, machine learning, and scientific measurement. In Bayesian parameter estimation, the posterior distribution contains uncertainty about an unknown quantity; posterior means, medians, intervals, and probabilities of scientifically relevant events can then be reported. Posterior predictive probabilities extend the same logic to future observations by averaging predictions over parameter values weighted by their posterior probabilities.4
Posterior probability can concern hypotheses that are not directly observable, but the result remains conditional on the chosen set of alternatives. If an important competing explanation is omitted, the normalized posterior probabilities may appear more decisive than they should be. This issue is sometimes called a reference-class or model-space problem.
Posterior odds provide another useful form: posterior odds equal prior odds multiplied by the Bayes factor, where the Bayes factor compares how well the evidence is predicted by two models. Unlike a p-value, a posterior probability directly assigns probability to a hypothesis under the model, though it does not remove the need to assess model adequacy.5 In hierarchical models, evidence can update both individual cases and population-level distributions. In sequential settings, exact Bayesian updating is order-invariant when the same evidence and model are used, but practical results can differ when analysts change the model, discard data, or treat dependent observations as independent.
Posterior probabilities are model-dependent quantities: their interpretation requires attention to the prior, likelihood, evidence selection, competing hypotheses, and dependence assumptions.
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