Conditional Independence for Statistical Operations

Abstract

A general calculus of conditional independence is developed, suitable for application to a wide range of statistical concepts such as sufficiency, parameter-identification, adequacy and ancillarity. A vehicle for this theory is the statistical operation, a structure-preserving map between statistical spaces. Concepts such as completeness and identifiability of mixtures arise naturally and play an important part. Some general theorems are exemplified by applications to ancillarity, including a study of a Bayesian definition of ancillarity in the presence of nuisance parameters.