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In decision theory, a decision rule is a function which maps an observation to an appropriate action. Decision rules play an important role in the theory of statistics and economics , and are closely related to the concept of a strategy in game theory .
In statistical decision theory, a randomised decision rule or mixed decision rule is a decision rule that associates probabilities with deterministic decision rules. In finite decision problems, randomised decision rules define a risk set which is the convex hull of the risk points of the nonrandomised decision rules.
In decision theory, a decision rule is said to dominate another if the performance of the former is sometimes better, and never worse, than that of the latter. Formally, let δ 1 {\displaystyle \delta _{1}} and δ 2 {\displaystyle \delta _{2}} be two decision rules , and let R ( θ , δ ) {\displaystyle R(\theta ,\delta )} be the risk of rule ...
A decision rule that minimizes (,) is called a Bayes rule with respect to (). There may be more than one such Bayes rule. There may be more than one such Bayes rule. If the Bayes risk is infinite for all δ {\displaystyle \delta \,\!} , then no Bayes rule is defined.
An estimator or decision rule with zero bias is called unbiased. In statistics, "bias" is an objective property of an estimator. Bias is a distinct concept from consistency : consistent estimators converge in probability to the true value of the parameter, but may be biased or unbiased (see bias versus consistency for more).
This optimal decision, a * is known as the Bayes (decision) Rule - it minimises the average loss over all possible states of nature θ, over all possible (probability-weighted) data outcomes. One advantage of the Bayesian approach is to that one need only choose the optimal action under the actual observed data to obtain a uniformly optimal one ...
The quadratic scoring rule is a strictly proper scoring rule (,) = = =where is the probability assigned to the correct answer and is the number of classes.. The Brier score, originally proposed by Glenn W. Brier in 1950, [4] can be obtained by an affine transform from the quadratic scoring rule.
Stein's example is surprising, since the "ordinary" decision rule is intuitive and commonly used. In fact, numerous methods for estimator construction, including maximum likelihood estimation, best linear unbiased estimation, least squares estimation and optimal equivariant estimation, all result in the "ordinary" estimator.