Part of a series on
Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
Background
Bayesian inference Bayesian probability Bayes' theorem Bernstein–von Mises theorem Coherence Cox's theorem Cromwell's rule Principle of indifference Principle of maximum entropy
Model building
Weak prior ... Strong prior Conjugate prior Linear regression Empirical Bayes Hierarchical model
Posterior approximation
Markov chain Monte Carlo Laplace's approximation Integrated nested Laplace approximations Variational inference Approximate Bayesian computation
Estimators
Bayesian estimator Credible interval Maximum a posteriori estimation
Evidence approximation
Evidence lower bound Nested sampling
Model evaluation
Bayes factor Model averaging Posterior predictive
v t e

Bayesian inference (/ˈbeɪziən/ BAY-zee-ən or /ˈbeɪʒən/ BAY-zhən)^[1] is a type of statistical inference. In Bayesian inference, evidence or information is available, Bayes' theorem is used to change (or update) the probability of a hypothesis. Bayesian inference uses a prior distribution to estimate posterior probabilities. Bayesian inference is an important to statistics, mathematical statistics, decision theory, and sequential analysis. Bayesian inference is used in science, engineering, philosophy, medicine, sport, and law.

Bayesian inference figures out the posterior probability from prior probability and the "likelihood function". The likelihood function comes from a statistical model of the data. $${\displaystyle P(H\mid E)={\frac {P(E\mid H)\cdot P(H)}{P(E))),}$$ where

• ${\displaystyle H}$ is a hypothesis that is changed by data (or evidence). There are usually many hypotheses. The point of the test is to see which hypothesis is more likely.
• ${\displaystyle P(H)}$ is the prior probability. It estimates the probability of a hypothesis before there is any evidence.
• ${\displaystyle E}$ is the evidence, or data. It is any new data that is found.
• ${\displaystyle P(H\mid E)}$ is the posterior probability. This is what we want to know.
• ${\displaystyle P(E\mid H)}$ is the likelihood function.
• ${\displaystyle P(E)}$ is the marginal likelihood. It is the same for all possible hypotheses that are being tested. ${\displaystyle P(E)}$ has to be greater than 0. If ${\displaystyle P(E)}$ is 0, then you divide by zero.

• Epistemology
• Free energy principle

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• Bayesian Statistics from Scholarpedia.
• Introduction to Bayesian probability from Queen Mary University of London
• Mathematical Notes on Bayesian Statistics and Markov Chain Monte Carlo
• Bayesian reading list Archived 2011-06-25 at the Wayback Machine, categorized and annotated by Tom Griffiths
• A. Hajek and S. Hartmann: Bayesian Epistemology, in: J. Dancy et al. (eds.), A Companion to Epistemology. Oxford: Blackwell 2010, 93–106.
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• Stanford Encyclopedia of Philosophy: "Inductive Logic"
• Bayesian Confirmation Theory (PDF)
• Data, Uncertainty and Inference — Informal introduction with many examples, ebook (PDF) freely available at causaScientia