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Admissible decision rule

In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is no other rule that is always "better" than it[1] (or at least sometimes better and never worse), in the precise sense of "better" defined below. This concept is analogous to Pareto efficiency.

Contents

DefinitionEdit

Define sets  ,   and  , where   are the states of nature,   the possible observations, and   the actions that may be taken. An observation   is distributed as   and therefore provides evidence about the state of nature  . A decision rule is a function  , where upon observing  , we choose to take action  .

Also define a loss function  , which specifies the loss we would incur by taking action   when the true state of nature is  . Usually we will take this action after observing data  , so that the loss will be  . (It is possible though unconventional to recast the following definitions in terms of a utility function, which is the negative of the loss.)

Define the risk function as the expectation

 

Whether a decision rule   has low risk depends on the true state of nature  . A decision rule   dominates a decision rule   if and only if   for all  , and the inequality is strict for some  .

A decision rule is admissible (with respect to the loss function) if and only if no other rule dominates it; otherwise it is inadmissible. Thus an admissible decision rule is a maximal element with respect to the above partial order. An inadmissible rule is not preferred (except for reasons of simplicity or computational efficiency), since by definition there is some other rule that will achieve equal or lower risk for all  . But just because a rule   is admissible does not mean it is a good rule to use. Being admissible means there is no other single rule that is always as good or better – but other admissible rules might achieve lower risk for most   that occur in practice. (The Bayes risk discussed below is a way of explicitly considering which   occur in practice.)

Bayes rules and generalized Bayes rulesEdit

Bayes rulesEdit

Let   be a probability distribution on the states of nature. From a Bayesian point of view, we would regard it as a prior distribution. That is, it is our believed probability distribution on the states of nature, prior to observing data. For a frequentist, it is merely a function on   with no such special interpretation. The Bayes risk of the decision rule   with respect to   is the expectation

 

A decision rule   that minimizes   is called a Bayes rule with respect to  . There may be more than one such Bayes rule. If the Bayes risk is infinite for all  , then no Bayes rule is defined.

Generalized Bayes rulesEdit

In the Bayesian approach to decision theory, the observed   is considered fixed. Whereas the frequentist approach (i.e., risk) averages over possible samples  , the Bayesian would fix the observed sample   and average over hypotheses  . Thus, the Bayesian approach is to consider for our observed   the expected loss

 

where the expectation is over the posterior of   given   (obtained from   and   using Bayes' theorem).

Having made explicit the expected loss for each given   separately, we can define a decision rule   by specifying for each   an action   that minimizes the expected loss. This is known as a generalized Bayes rule with respect to  . There may be more than one generalized Bayes rule, since there may be multiple choices of   that achieve the same expected loss.

At first, this may appear rather different from the Bayes rule approach of the previous section, not a generalization. However, notice that the Bayes risk already averages over   in Bayesian fashion, and the Bayes risk may be recovered as the expectation over   of the expected loss (where   and  ). Roughly speaking,   minimizes this expectation of expected loss (i.e., is a Bayes rule) if and only if it minimizes the expected loss for each   separately (i.e., is a generalized Bayes rule).

Then why is the notion of generalized Bayes rule an improvement? It is indeed equivalent to the notion of Bayes rule when a Bayes rule exists and all   have positive probability. However, no Bayes rule exists if the Bayes risk is infinite (for all  ). In this case it is still useful to define a generalized Bayes rule  , which at least chooses a minimum-expected-loss action   for those   for which a finite-expected-loss action does exist. In addition, a generalized Bayes rule may be desirable because it must choose a minimum-expected-loss action   for every  , whereas a Bayes rule would be allowed to deviate from this policy on a set   of measure 0 without affecting the Bayes risk.

More important, it is sometimes convenient to use an improper prior  . In this case, the Bayes risk is not even well-defined, nor is there any well-defined distribution over  . However, the posterior  —and hence the expected loss—may be well-defined for each  , so that it is still possible to define a generalized Bayes rule.

Admissibility of (generalized) Bayes rulesEdit

According to the complete class theorems, under mild conditions every admissible rule is a (generalized) Bayes rule (with respect to some prior  —possibly an improper one—that favors distributions   where that rule achieves low risk). Thus, in frequentist decision theory it is sufficient to consider only (generalized) Bayes rules.

Conversely, while Bayes rules with respect to proper priors are virtually always admissible, generalized Bayes rules corresponding to improper priors need not yield admissible procedures. Stein's example is one such famous situation.

ExamplesEdit

The James–Stein estimator is a nonlinear estimator of the mean of Gaussian random vectors which can be shown to dominate, or outperform, the ordinary least squares technique with respect to a mean-square error loss function.[2] Thus least squares estimation is not an admissible estimation procedure in this context. Some others of the standard estimates associated with the normal distribution are also inadmissible: for example, the sample estimate of the variance when the population mean and variance are unknown.[3]

NotesEdit

  1. ^ Dodge, Y. (2003) The Oxford Dictionary of Statistical Terms. OUP. ISBN 0-19-920613-9 (entry for admissible decision function)
  2. ^ Cox & Hinkley 1974, Section 11.8
  3. ^ Cox & Hinkley 1974, Exercise 11.7

ReferencesEdit

  • Cox, D. R.; Hinkley, D. V. (1974). Theoretical Statistics. Wiley. ISBN 0-412-12420-3.
  • Berger, James O. (1980). Statistical Decision Theory and Bayesian Analysis (2nd ed.). Springer-Verlag. ISBN 0-387-96098-8.
  • DeGroot, Morris (2004) [1st. pub. 1970]. Optimal Statistical Decisions. Wiley Classics Library. ISBN 0-471-68029-X.
  • Robert, Christian P. (1994). The Bayesian Choice. Springer-Verlag. ISBN 3-540-94296-3.