Bayesian inference is a method of statistical inference in which some kinds of evidence or observations are used to calculate the probability that a hypothesis may be true, or else to update its previously calculated probability. The term "Bayesian" comes from its use of the Bayes' theorem in the calculation process. (Bayes' theorem was deduced in several special cases by Thomas Bayes, and then it was extended to the general theorem by other researchers.)
In practical usage, "Bayesian inference" refers to the use of a prior probability over hypotheses to determine the probability of a particular hypothesis given some observed evidence; that is, the probability that a particular hypothesis is true given some observed evidence (the posterior probability of the hypothesis) comes from a combination of the prior probability of the hypothesis and the compatibility of the observed evidence with the hypothesis (or likelihood of the evidence, in a technical sense). Bayesian inference is different from frequentist inference, which uses the sampling distribution of a statistic. Most elementary undergraduate-level statistics courses teach frequentist inference rather than Bayesian inference.
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