Bayesian statistics

Bayesian statistics
Theory
Bayesian probability Probability interpretations Bayes' theorem Bayes' rule Bayes factor Bayesian inference Bayesian network Prior Posterior Likelihood Conjugate prior Posterior predictive Hyperparameter Hyperprior Principle of indifference Principle of maximum entropy Empirical Bayes method Cromwell's rule Bernstein–von Mises theorem Bayesian information criterion Credible interval Maximum a posteriori estimation
Techniques
Bayesian linear regression Bayesian estimator Approximate Bayesian computation
Uses
Bayesian spam filtering Binary classification Naive Bayes classifier
Statistics portal

Bayesian statistics is a subset of the 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".

Outline

The general set of statistical techniques can be divided into a number of activities, many of which have special "Bayesian" versions.

Statistical inference

Main article: Bayesian inference

Bayesian inference is an approach to statistical inference, that is distinct from frequentist inference. It is specifically based on the use of Bayesian probabilities to summarise evidence.

Statistical modelling

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 allow 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

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.

External links

• Theo Kypraios. "A Gentle Tutorial in Bayesian Statistics" (PDF). Retrieved 2012-09-22. 
• Bayesian modeling book and examples available for downloading.
• Bayesian statistics at Wikiversity