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Bayesian statistics
Posterior = Likelihood × Prior ÷ Evidence
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Bayes' theorem
Bernstein–von Mises theorem
Coherence
Cox's theorem
Cromwell's rule
Principle of indifference
Principle of maximum entropy
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Weak prior ... Strong prior
Conjugate prior
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Interpretation of probability
Bayesian probability (/ˈbeɪziən/BAY-zee-ən or /ˈbeɪʒən/BAY-zhən)[1] is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation[2] representing a state of knowledge[3] or as quantification of a personal belief.[4]
The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses;[5][6] that is, with propositions whose truth or falsity is unknown. In the Bayesian view, a probability is assigned to a hypothesis, whereas under frequentist inference, a hypothesis is typically tested without being assigned a probability.
Bayesian probability belongs to the category of evidential probabilities; to evaluate the probability of a hypothesis, the Bayesian probabilist specifies a prior probability. This, in turn, is then updated to a posterior probability in the light of new, relevant data (evidence).[7] The Bayesian interpretation provides a standard set of procedures and formulae to perform this calculation.
The term Bayesian derives from the 18th-century mathematician and theologian Thomas Bayes, who provided the first mathematical treatment of a non-trivial problem of statistical data analysis using what is now known as Bayesian inference.[8]: 131 Mathematician Pierre-Simon Laplace pioneered and popularized what is now called Bayesian probability.[8]: 97–98
^"Bayesian". Merriam-Webster.com Dictionary.
^Cox, R.T. (1946). "Probability, Frequency, and Reasonable Expectation". American Journal of Physics. 14 (1): 1–10. Bibcode:1946AmJPh..14....1C. doi:10.1119/1.1990764.
^Jaynes, E.T. (1986). "Bayesian Methods: General Background". In Justice, J. H. (ed.). Maximum-Entropy and Bayesian Methods in Applied Statistics. Cambridge: Cambridge University Press. CiteSeerX 10.1.1.41.1055.
^de Finetti, Bruno (2017). Theory of Probability: A critical introductory treatment. Chichester: John Wiley & Sons Ltd. ISBN 9781119286370.
^Hailperin, Theodore (1996). Sentential Probability Logic: Origins, Development, Current Status, and Technical Applications. London: Associated University Presses. ISBN 0934223459.
^Howson, Colin (2001). "The Logic of Bayesian Probability". In Corfield, D.; Williamson, J. (eds.). Foundations of Bayesianism. Dordrecht: Kluwer. pp. 137–159. ISBN 1-4020-0223-8.
^Paulos, John Allen (5 August 2011). "The Mathematics of Changing Your Mind [by Sharon Bertsch McGrayne]". Book Review. New York Times. Archived from the original on 2022-01-01. Retrieved 2011-08-06.
^ abStigler, Stephen M. (March 1990). The history of statistics. Harvard University Press. ISBN 9780674403413.
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