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. As nonrandomised alternatives always exist to randomised Bayes rules, randomisation is not needed in Bayesian statistics, although frequentist statistical theory sometimes requires the use of randomised rules to satisfy optimality conditions such as minimax, most notably when deriving confidence intervals and hypothesis tests about discrete probability distributions. A statistical test making use of a randomized decision rule is called a randomized test.
Definition and interpretation Let D = { d 1 , d 2 . . . , d h } {\displaystyle {\mathcal {D}}=\{d_{1},d_{2}...,d_{h}\}} be a set of non-randomised decision rules with associated probabilities p 1 , p 2 , . . . , p h {\displaystyle p_{1},p_{2},...,p_{h}} . Then the randomised decision rule d ∗ {\displaystyle d^{*}} is defined as ∑ i = 1 h p i d i {\displaystyle \sum _{i=1}^{h}p_{i}d_{i}} and its associated risk function R ( θ , d ∗ ) {\displaystyle R(\theta ,d^{*})} is ∑ i = 1 h p i R ( θ , d i ) {\displaystyle \sum _{i=1}^{h}p_{i}R(\theta ,d_{i})} . This rule can be treated as a random experiment in which the decision rules d 1 , . . . , d h ∈ D {\displaystyle d_{1},...,d_{h}\in {\mathcal {D}}} are selected with probabilities p 1 , . . . p h {\displaystyle p_{1},...p_{h}} respectively. Alternatively, a randomised decision rule may assign probabilities directly on elements of the actions space A {\displaystyle {\mathcal {A}}} for each member of the sample space. More formally, d ∗ ( x , A ) {\displaystyle d^{*}(x,A)} denotes the probability that an action a ∈ A {\displaystyle a\in {\mathcal {A}}} is chosen. Under this approach, its loss function is also defined directly as: ∫ A ∈ A d ∗ ( x , A ) L ( θ , A ) d A {\displaystyle \int _{A\in {\mathcal {A}}}d^{*}(x,A)L(\theta ,A)dA} . The introduction of randomised decision rules thus creates a larger decision space from which the statistician may choose his decision. As non-randomised decision rules are a special case of randomised decision rules where one decision or action has probability 1, the original decision space D {\displaystyle {\mathcal {D}}} is a proper subset of the new decision space D ∗ {\displaystyle {\mathcal {D}}^{*}} .
Selection of randomised decision rules
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