The paper presents, from a Bayesian point of view, several results about conditional independence and about its role in analyzing the relations between concepts of sufficiency and concepts of invariance. Some of the given results are, in particular, related with the Bayesian translation of the “Stein theorem” [see W.J. Hall, R.A. Wijsman and J.K. Ghosh, Ann. Math. Stat. 36, 575-614 (1965; Zbl 0227.62007)], which gives sufficient conditions under which is sufficient for , where is a sufficient -field and is the -field of all events, invariant under the action of a given group .
The paper starts with a brief review about concepts of sufficiency and invariance (both in the classic and Bayesian statistics settings) and ends with some further discussion about the Stein theorem. Some examples of the theory are provided.