The Minimax Rule In The Multi-Hypothesis Bayes Decision

In statistical decision theory, there exist quite few rules which are depen-dent on different information we have known. For example, Bayes rule, Neyman-Pearson rule, MAP rule, Sequential rule, etc.We must know the prior proba-bilities when we use all of the rules except Neyman-Pearson rule. But Neyman-Pearson rule is only suitable for binary decision. When using the Bayes rule, weneed to know the cost coefficient, prior probability of the hypotheses and thejoint conditional probability density functions. When the prior probabilities areunknown, we can use the minimax rule to get Bayes decision of the worst priorprobabilities in binary decision(see [14]). This thesis mainly extend the minimaxrule to general multi-hypothesis decision, in which a group of minimax equa-tions arederived, the corresponding algorithm to solve the minimax equationsare presented. When the function forms of prior probability density functionsare clear, but the some parameters are unknown, literature [23] have given acomposite decision method. We also discuss the minimax decision rule whenthe prior probabilities and the parameters of prior probability density functionsare unknown, we also give three numerical examples which show the methodand algorithm are efficient. At the end, We discuss how to estimate the priorprobabilities when we can receive some independent observation, The numericalexamples show the method and algorithm are convergent.