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Bayesian inference

From Simple English Wikipedia, the free encyclopedia

Bayesian inference (/ˈbziən/ BAY-zee-ən or /ˈbʒən/ BAY-zhən)[1] is a type of statistical inference. In Bayesian inference, evidence or information is available, Bayes' theorem is used to change (or update) the probability of a hypothesis. Bayesian inference uses a prior distribution to estimate posterior probabilities. Bayesian inference is an important to statistics, mathematical statistics, decision theory, and sequential analysis. Bayesian inference is used in science, engineering, philosophy, medicine, sport, and law.

Bayes' rule[change]

File:Bayes theorem visualisation.svg
A geometric visualisation of Bayes' theorem. In the table, the values 2, 3, 6 and 9 give the relative weights of each corresponding condition and case. The figures denote the cells of the table involved in each metric, the probability being the fraction of each figure that is shaded. This shows that P(A|B) P(B) = P(B|A) P(A) i.e. P(A|B) = P(B|A) P(A)/P(B) . Similar reasoning can be used to show that P(¬A|B) = P(B|¬A) P(¬A)/P(B) etc.
Contingency table
Hypothesis


Evidence
Satisfies
hypothesis
H
Violates
hypothesis
¬H

Total
Has evidence
E
P(H|E)·P(E)
= P(E|H)·P(H)
P(¬H|E)·P(E)
= P(E|¬H)·P(¬H)
P(E)
No evidence
¬E
P(H|¬E)·P(¬E)
= P(¬E|H)·P(H)
P(¬H|¬E)·P(¬E)
= P(¬E|¬H)·P(¬H)
P(¬E) =
1−P(E)
Total    P(H) P(¬H) = 1−P(H) 1

Bayesian inference figures out the posterior probability from prior probability and the "likelihood function". The likelihood function comes from a statistical model of the data. <math display="block">P(H \mid E) = \frac{P(E \mid H) \cdot P(H)}{P(E)},</math> where

Related pages[change]

Further reading[change]

  • Vallverdu, Jordi (2016). Bayesians Versus Frequentists A Philosophical Debate on Statistical Reasoning. New York: Springer. ISBN 978-3-662-48638-2.
  • Clayton, Aubrey (August 2021). Bernoulli's Fallacy: Statistical Illogic and the Crisis of Modern Science. Columbia University Press. ISBN 978-0-231-55335-3.

References[change]

Other websites[change]