Posterior probability

A probability distribution over hypotheses or parameters after observed evidence updates a prior through Bayes’ theorem. It represents uncertainty conditional on the evidence.

Bayes’ theorem: A rule relating conditional probabilities that updates a hypothesis’s probability using observed evidence. It computes the posterior by combining the prior with the evidence’s likelihood.
Bayesian inference: Statistical inference that represents uncertainty with probability distributions and updates them using Bayes’ theorem. Posterior distributions are the central output of Bayesian inference.
Bayesian model comparison: The comparison of statistical models using their posterior probabilities or marginal likelihoods. Model probabilities are posterior probabilities over competing explanations.
Likelihood: A function of model parameters proportional to the probability of observed data under those parameters. Likelihood evaluates data under fixed possibilities; posterior probability also incorporates prior information.
Prior probability: A probability distribution expressing uncertainty about hypotheses or parameters before incorporating current evidence. It supplies the starting distribution that evidence updates.
Probability distribution: A mathematical description assigning probabilities or probability densities to possible outcomes or parameter values. A posterior over a continuous parameter is a distribution, not a probability for each exact value.
Bayesian decision theory: A framework for choosing actions by combining probabilities with the consequences of outcomes. Decisions can use posterior probabilities to evaluate expected losses or benefits.
P-value: The probability, under a specified null model, of data at least as extreme as those observed. A p-value is not the probability that a hypothesis is true after seeing data.
Likelihood function: A function measuring how probable observed data are under different parameter values or hypotheses. It determines which possibilities the evidence supports more strongly.
Hypothesis testing: A statistical procedure for assessing evidence about specified hypotheses using a defined decision rule. Posterior probabilities quantify hypotheses directly, unlike many classical testing procedures.

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