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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 (or at least sometimes better and never worse), in the precise sense of "better" defined below. This concept is analogous to
Pareto efficiency In welfare economics, a Pareto improvement formalizes the idea of an outcome being "better in every possible way". A change is called a Pareto improvement if it leaves at least one person in society better off without leaving anyone else worse ...
.


Definition

Define sets \Theta\,, \mathcal and \mathcal, where \Theta\, are the states of nature, \mathcal the possible observations, and \mathcal the actions that may be taken. An observation of x \in \mathcal\,\! is distributed as F(x\mid\theta)\,\! and therefore provides evidence about the state of nature \theta\in\Theta\,\!. A decision rule is a function \delta:\rightarrow , where upon observing x\in \mathcal, we choose to take action \delta(x)\in \mathcal\,\!. Also define a loss function L: \Theta \times \mathcal \rightarrow \mathbb, which specifies the loss we would incur by taking action a \in \mathcal when the true state of nature is \theta \in \Theta. Usually we will take this action after observing data x \in \mathcal, so that the loss will be L(\theta,\delta(x))\,\!. (It is possible though unconventional to recast the following definitions in terms of a
utility function In economics, utility is a measure of a certain person's satisfaction from a certain state of the world. Over time, the term has been used with at least two meanings. * In a Normative economics, normative context, utility refers to a goal or ob ...
, which is the negative of the loss.) Define the risk function as the expectation :R(\theta,\delta)=\operatorname_ dominates a decision rule \delta\,\! if and only if R(\theta,\delta^*)\le R(\theta,\delta) for all \theta\,\!, ''and'' the inequality is inequality (mathematics)">strict In mathematical writing, the term strict refers to the property of excluding equality and equivalence and often occurs in the context of inequality and monotonic functions. It is often attached to a technical term to indicate that the exclusiv ...
for some \theta\,\!. 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'' \theta\,\!. But just because a rule \delta\,\! 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 \theta\,\! that occur in practice. (The Bayes risk discussed below is a way of explicitly considering which \theta\,\! occur in practice.)


Bayes rules and generalized Bayes rules


Bayes rules

Let \pi(\theta)\,\! be a probability distribution on the states of nature. From a Bayesian point of view, we would regard it as a ''
prior distribution A prior probability distribution of an uncertain quantity, simply called the prior, is its assumed probability distribution before some evidence is taken into account. For example, the prior could be the probability distribution representing the ...
''. 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 \Theta\,\! with no such special interpretation. The Bayes risk of the decision rule \delta\,\! with respect to \pi(\theta)\,\! is the expectation :r(\pi,\delta)=\operatorname_ (\theta,\delta)\,\! A decision rule \delta\,\! that minimizes r(\pi,\delta)\,\! is called a Bayes rule with respect to \pi(\theta)\,\!. There may be more than one such Bayes rule. If the Bayes risk is infinite for all \delta\,\!, then no Bayes rule is defined.


Generalized Bayes rules

In the Bayesian approach to decision theory, the observed x\,\! is considered ''fixed''. Whereas the frequentist approach (i.e., risk) averages over possible samples x \in \mathcal\,\!, the Bayesian would fix the observed sample x\,\! and average over hypotheses \theta \in \Theta\,\!. Thus, the Bayesian approach is to consider for our observed x\,\! the expected loss :\rho(\pi,\delta \mid x)=\operatorname_ L(\theta,\delta(x)) \,\! where the expectation is over the ''posterior'' of \theta\,\! given x\,\! (obtained from \pi(\theta)\,\! and F(x\mid\theta)\,\! using
Bayes' theorem Bayes' theorem (alternatively Bayes' law or Bayes' rule, after Thomas Bayes) gives a mathematical rule for inverting Conditional probability, conditional probabilities, allowing one to find the probability of a cause given its effect. For exampl ...
). Having made explicit the expected loss for each given x\,\! separately, we can define a decision rule \delta\,\! by specifying for each x\,\! an action \delta(x)\,\! that minimizes the expected loss. This is known as a generalized Bayes rule with respect to \pi(\theta)\,\!. There may be more than one generalized Bayes rule, since there may be multiple choices of \delta(x)\,\! 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 \Theta\,\! in Bayesian fashion, and the Bayes risk may be recovered as the expectation over \mathcal of the expected loss (where x\sim\theta\,\! and \theta\sim\pi\,\!). Roughly speaking, \delta\,\! minimizes this expectation of expected loss (i.e., is a Bayes rule) if and only if it minimizes the expected loss for each x \in \mathcal 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 x\,\! have positive probability. However, no Bayes rule exists if the Bayes risk is infinite (for all \delta\,\!). In this case it is still useful to define a generalized Bayes rule \delta\,\!, which at least chooses a minimum-expected-loss action \delta(x)\!\, for those x\,\! 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 \delta(x)\,\! for ''every'' x\,\!, whereas a Bayes rule would be allowed to deviate from this policy on a set X \subseteq \mathcal of measure 0 without affecting the Bayes risk. More important, it is sometimes convenient to use an improper prior \pi(\theta)\,\!. In this case, the Bayes risk is not even well-defined, nor is there any well-defined distribution over x\,\!. However, the posterior \pi(\theta\mid x)\,\!—and hence the expected loss—may be well-defined for each x\,\!, so that it is still possible to define a generalized Bayes rule.


Admissibility of (generalized) Bayes rules

According to the complete class theorems, under mild conditions every admissible rule is a (generalized) Bayes rule (with respect to some prior \pi(\theta)\,\!—possibly an improper one—that favors distributions \theta\,\! where that rule achieves low risk). Thus, in frequentist
decision theory Decision theory or the theory of rational choice is a branch of probability theory, probability, economics, and analytic philosophy that uses expected utility and probabilities, probability to model how individuals would behave Rationality, ratio ...
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.


Examples

The James–Stein estimator is a nonlinear estimator of the mean of Gaussian random vectors and can be shown to dominate the
ordinary least squares In statistics, ordinary least squares (OLS) is a type of linear least squares method for choosing the unknown parameters in a linear regression In statistics, linear regression is a statistical model, model that estimates the relationship ...
technique with respect to a mean-squared-error loss function. Thus least squares estimation is not an admissible estimation procedure in this context. Some others of the standard estimates associated with the
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
are also inadmissible: for example, the sample estimate of the variance when the population mean and variance are unknown.


Notes


References

* * * *{{cite book , author=Robert, Christian P. , title=The Bayesian Choice , publisher=Springer-Verlag , year=1994 , isbn=3-540-94296-3 Bayesian statistics Optimal decisions