Editing Bayesian inference (section)

===General formulation===
[[File:Bayesian inference event space.svg|thumb|Diagram illustrating event space <math>\Omega</math> in general formulation of Bayesian inference. Although this diagram shows discrete models and events, the continuous case may be visualized similarly using probability densities.]]

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Suppose a process is generating independent and identically distributed events <math>E_n,\ n = 1, 2, 3, \ldots</math>, but the [[probability distribution]] is unknown. Let the event space <math>\Omega</math> represent the current state of belief for this process. Each model is represented by event <math>M_m</math>. The conditional probabilities <math>P(E_n \mid M_m)</math> are specified to define the models. <math>P(M_m)</math> is the [[Credence (statistics)|degree of belief]] in <math>M_m</math>. Before the first inference step, <math>\{P(M_m)\}</math> is a set of ''initial prior probabilities''. These must sum to 1, but are otherwise arbitrary.

Suppose that the process is observed to generate <math>E \in \{E_n\}</math>. For each <math>M \in \{M_m\}</math>, the prior <math>P(M)</math> is updated to the posterior <math>P(M \mid E)</math>. From [[Bayes' theorem]]:<ref>Gelman, Andrew; Carlin, John B.; Stern, Hal S.; Dunson, David B.; Vehtari, Aki; Rubin, Donald B. (2013). ''Bayesian Data Analysis'', Third Edition. Chapman and Hall/CRC. {{ISBN|978-1-4398-4095-5}}.</ref>

<math display="block">P(M \mid E) = \frac{P(E \mid M)}{\sum_m {P(E \mid M_m) P(M_m)}} \cdot P(M).</math>

Upon observation of further evidence, this procedure may be repeated.