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c1s3 - 1.3 Bayes decision theory The distinguishing feature...

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March 18, 2003 1.3 Bayes decision theory . The distinguishing feature of Bayesian statistics is that a probability distribution π , called a prior , is given on the parameter space , T ). Some- times, priors are also considered which may be infinite, such as Lebesgue measure on the whole real line, but such priors will not be treated here at least for the time being. A Bayesian statistician chooses a prior π based on whatever information on the un- known θ is available in advance of making any observations in the current experiment. In general, no definite rules are prescribed for choosing π . Priors are often useful as technical tools in reaching non-Bayesian conclusions such as admissibility in Theorems 1.2.5 and 1.2.6. Bayes decision rules were defined near the end of the last section as rules which minimize the Bayes risk and for which the risk is finite. Bayes tests of P vs. Q , treated in Theorem 1.1.8, are a special case of Bayes decision rules. We saw in that case that Bayes rules need not be randomized (Remark 1.1.9). The same is true quite generally in Bayes decision theory: if, in a given situation, it is Bayes to choose at random among two or more possible decisions, then the decisions must have equal risks (conditional on the observations) and we may as well just take one of them. Theorem 1.3.1 will give a more precise statement. In game theory, randomization is needed to have a strategy that is optimal even if the opponent knows it and can choose a strategy accordingly. If one knows the opponent’s strategy then it is not necessary to randomize. Sometimes, statistical decision theory is viewed as a game against an opponent called “Nature.” Unlike an opponent in game theory, “Nature” is viewed as neutral, not trying to win the game. Assuming a prior, as in Bayes decision theory, is to assume in effect that “Nature” follows a certain strategy. In showing that randomization isn’t needed, it will be helpful to formulate randomiza- tion in a fuller way, where we not only choose a probability distribution over the possible actions, but then also choose an action according to that distribution, in a measurable way, as follows: Definition . A randomized decision rule d : X D E is realizable if there is a probability space (Ω , F , µ ) and a jointly measurable function δ : X × A such that for each x in X , δ ( x, · ) has distribution d ( x ), in other words d ( x ) is the image measure of µ by δ ( x, · ) , d ( x ) = µ δ ( x, · ) 1 . For example, a randomized test as in Sec. 1.1 is always a realizable rule, where we can take as the interval [0 , 1] with Lebesgue measure and let δ ( x, t ) = d Q if t f ( x ) and d P otherwise. It is shown in the next section that decision rules are realizable under conditions wide enough to cover a great many cases, for example whenever the action space is a subset of a space R k with Borel σ -algebra. It will be shown next that randomization is unnecessary for realizable Bayes rules. The idea is that the Bayes risk of a realizable randomized Bayes rule d ( · ) is an average of Bayes risks of
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