Bayesian statistics is that subset of the entire field of statistics in which the evidence about the true state of the world is expressed in terms of degrees of belief or, more specifically, Bayesian probabilities. Such an interpretation is only one of a number of interpretations of probability and there are many other statistical techniques that are not based on "degrees of belief".
The general set of statistical techniques can be divided into a number of activities, many of which have special "Bayesian" versions.
Main article: Bayesian inference
Bayesian inference is an approach to statistical inference, that is distinct from the more traditional frequentist inference. It is specifically based on the use of Bayesian probabilities to summarise evidence.
The formulation of statistical models for use in Bayesian statistics has the additional feature, not present with other types of statistical techniques, of requiring the formulation of a set of prior distributions for any unknown parameters. Such prior distributions are as much part of the statistical model as the part that expresses the probability distribution of observations given the model parameters. The specification of a set of prior distributions for a problem may involve hyperparameters and hyperprior distributions.
Design of experiments
Main article: Bayesian design of experiments
The usual considerations in the design of experiments are extended in the case of Bayesian design of experiments to include the influence of prior beliefs. Importantly, the application of sequential analysis techniques allows the outcome of earlier experiments to influence the design of the next experiment, based on the updating of beliefs as expressed by the prior and posterior distribution. Part of the problem of the design of experiments is that they should make good use of resources of all types: one example of the Bayesian design of experiments aimed at such efficiency is the multi-armed bandit problem.
Statistical graphics includes methods for data exploration, for model validation, etc. The use of certain modern computational techniques for Bayesian inference, specifically the various types of Markov chain Monte Carlo techniques, have led to the need for checks, often made in graphical form, on the validity of such computations in expressing the required posterior distributions.
Bayesian modeling book and examples available for downloading.
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