Pattern Recognition, 4th Edition

Appendix A. Hints from Probability and Statistics

A.1. Total Probability and the Bayes Rule

Let Aⁱ, i = 1, 2,…, M, be M events so that ∑^{Mi = 1}P(Aⁱ) = 1 P(Aⁱ) = 1. Then the probability of an arbitrary event B is given by(A.1)where P(B|A) denotes the conditional probability of B assuming A, which is defined as(A.2)and P(B, A) is the joint probability of the two events. Equation (A.1) is known as the total probability theorem. From the definition in (A.2) the Bayes rule is readily available(A.3)These are easily extended to random variables or vectors ...