Bayesian statistics

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Bayesian statistics
Theory
Admissible decision rule Bayesian efficiency 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
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Bayesian statistics, named for Thomas Bayes (1701–1761), is a theory in the field of statistics in which the evidence about the true state of the world is expressed in terms of degrees of belief known as Bayesian probabilities. Such an interpretation is only one of a number of interpretations of probability and there are other statistical techniques that are not based on 'degrees of belief'. One of the key ideas of Bayesian statistics is that "probability is orderly opinion, and that inference from data is nothing other than the revision of such opinion in the light of relevant new information."^[1]

Outline

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

Statistical 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 summarize evidence.

Statistical modeling

The formulation of statistical models using Bayesian statistics has the unique feature of requiring the specification of prior distributions for any unknown parameters. These prior distributions are as integral to a Bayesian approach to statistical modelling as the expression of probability distributions. Prior distributions can be either hyperparameters or hyperprior distributions.

Design of experiments

The Bayesian design of experiments includes a concept called 'influence of prior beliefs'. This approach uses sequential analysis techniques to include the outcome of earlier experiments in the design of the next experiment. This is achieved by updating 'beliefs' through the use of prior and posterior distribution. This allows the design of experiments to make good use of resources of all types. An example of this 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.

References

Further reading

• Think Bayes, Allen B. Downey
• Bayesian Statistics: Why and How
• Puga JL, Krzywinski M, Altman N (May 2015). "Bayesian Statistics". Points of Significance. Nature Methods. 12 (5): 377–8. doi:10.1038/nmeth.3368. Retrieved 31 May 2016. 

External links

Wikiversity has learning materials about Bayesian statistics

• Eliezer S. Yudkowsky. "An Intuitive Explanation of Bayes' Theorem" (webpage). Retrieved 2015-06-15. 
• Theo Kypraios. "A Gentle Tutorial in Bayesian Statistics" (PDF) (PDF). Retrieved 2013-11-03. 
• Theo Kypraios. "Introduction to Bayesian Statistics" (PDF) (PDF). Retrieved 2014-05-05. 
• Jordi Vallverdu. "Bayesians Versus Frequentists A Philosophical Debate on Statistical Reasoning". Retrieved 2015-12-17. 
• Bayesian statistics David Spiegelhalter, Kenneth Rice Scholarpedia 4(8):5230. doi:10.4249/scholarpedia.5230
• Bayesian modeling book and examples available for downloading.
• Rens Van De Schoot. "A Gentle Introduction to Bayesian Analysis" (PDF).