Admissible decision rule

In statistical decision theory, an admissible decision rule is a rule for making a decision such that there is not any other rule that is always "better" than it.

Generally speaking, in most decision problems the set of admissible rules is large, even infinite, so this is not a sufficient criterion to pin down a single rule, but as will be seen there are some good reasons to favor admissible rules; compare Pareto efficiency.

Define sets ${\displaystyle \Theta \,}$, ${\displaystyle {\mathcal {X}}}$ and ${\displaystyle {\mathcal {A}}}$, where ${\displaystyle \Theta \,}$ are the states of nature, ${\displaystyle {\mathcal {X}}}$ the possible observations, and ${\displaystyle {\mathcal {A}}}$ the actions that may be taken. An observation ${\displaystyle x\in {\mathcal {X}}\,\!}$ is distributed as ${\displaystyle F(x\mid \theta )\,\!}$ and therefore provides evidence about the state of nature ${\displaystyle \theta \in \Theta \,\!}$. A decision rule is a function ${\displaystyle \delta :{\mathcal {X}}\rightarrow {\mathcal {A}}}$, where upon observing ${\displaystyle x\in {\mathcal {X}}}$, we choose to take action ${\displaystyle \delta (x)\in {\mathcal {A}}\,\!}$.

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