ABSTRACT
This chapter formalizes the idea that a given reduction of a Bayesian experiment “does not lose useful information”. This concept leads to the definition of admissible reductions. A parameter is said to be sufficient if the corresponding marginal reduction is admissible. A parameter is said to be ancillary if the corresponding conditional reduction is admissible. In a Bayesian framework, those concepts are naturally rendered operational by using the tool of stochastic independence both in marginal and in conditional terms. The concepts and equalities discussed are stated in terms of positive real-valued functions and are extended to integrable functions using standard arguments. The chapter discusses conditional independence in terms of densities. The basic idea is to start with a probability for which conditional independence is easily verified and to then characterize, in terms of the corresponding densities, conditional independence with respect to another probability dominated by the first one.
