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2.4.2 Bayes Theorem

Knowledge Base > Math> Probability Theory

Overview

The computation of a posterior probability P(Aj|B) from given prior probabilities P(Ai) and conditional probabilities P(B|Ai) can be formed in a theorem. Let A1,…,Ak be mutually exclusive and exhaustive events. Then for any other event B,

P (B ) = P (B |A1 )⋅P(A1)+ ...+ P(B |Ak )⋅P(Ak) ∑k = P (B |Ai) ⋅P(Ai) (2.4.7) i=1


Let A1,A2,…,Ak be a collection of k mutually exclusive and exhaustive events with P(Ai) > 0 for i = 1,…,k. Then for any other event B for which P(B) > 0
⋂ P (Aj |B ) = P(Aj---B) P (B ) = ∑-P-(B-|Aj)⋅P(Aj)---,∀j = 1,...,k (2.4.8) ki=1 P(B |Ai)⋅P (Ai)

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