What is the practical use of a complete-class theorem?

It tells you that you can confine your search for a good rule to the Bayes rules and their limits without missing anything admissible, which simplifies both theory and the construction of procedures.

Does this mean every good rule is Bayesian?

Essentially yes within the decision-theoretic framework: under the standard conditions every admissible rule is a Bayes rule or a limit of them, though the relevant prior may be improper or only emerge as a limit.

Complete-Class Theorems

A complete class is a set of decision rules rich enough that nothing outside it is worth using; complete-class theorems identify such sets with the Bayes rules and their limits.

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Definition

A class of decision rules is complete if for every rule outside it there is a rule inside it with uniformly no greater risk; complete-class theorems show that the admissible rules essentially coincide with the Bayes rules and their limits.

Scope

This topic covers complete and essentially complete classes of decision rules, the convexity and compactness of the risk set that drive the theory, the result that every admissible rule is Bayes or a limit of Bayes rules, the converse that Bayes rules under mild conditions are admissible, Wald's complete-class theorem and Stein's necessary-and-sufficient conditions, and the practical consequence of restricting attention to Bayes rules.

Core questions

What distinguishes a complete class from an essentially complete class?
Why does convexity of the risk set make Bayes rules central?
In what sense is every admissible rule a Bayes or limiting-Bayes rule?
How do complete-class theorems justify restricting attention to Bayes rules?

Key theories

Bayes characterization of admissibility: Under convexity and compactness conditions on the risk set, the class of Bayes rules and their limits is complete, so every admissible rule is Bayes or a limit of Bayes rules.
Wald and Stein complete-class theorems: Wald established the first complete-class results for statistical games, and Stein gave necessary and sufficient conditions for a class to be complete, sharpening the link between admissibility and Bayes optimality.

Clinical relevance

Complete-class theorems give a frequentist justification for Bayesian procedures: because the admissible rules are essentially the Bayes rules, searching among Bayes rules loses nothing, which is why Bayes and regularized estimators are reasonable defaults even under non-Bayesian criteria.

History

Wald proved the first complete-class theorems in his 1950 book on statistical decision functions. Blackwell, Stein, and Le Cam refined the conditions through the 1950s, establishing the now-standard equivalence between admissibility and Bayes optimality.

Key figures

Abraham Wald
Charles Stein
David Blackwell
James O. Berger

Seminal works

berger1985

Frequently asked questions

What is the practical use of a complete-class theorem?: It tells you that you can confine your search for a good rule to the Bayes rules and their limits without missing anything admissible, which simplifies both theory and the construction of procedures.
Does this mean every good rule is Bayesian?: Essentially yes within the decision-theoretic framework: under the standard conditions every admissible rule is a Bayes rule or a limit of them, though the relevant prior may be improper or only emerge as a limit.