Remarks on Bayes sufficiency (Q749062)

It is well known that there are various definitions of ``sufficiency''. In addition to the usual definition of sufficiency represented by the existence of a common conditional probability, we have the notions of pairwise sufficiency, PSS (pairwise sufficiency with supports), test sufficiency and Bayes sufficiency. These notions coincide with one another in the dominated case. In this note we show, by examples, that PSS does not imply Bayes sufficiency in case of a continuous a priori distribution, and the existence of the smallest PSS which is Bayes sufficient. This latter example shows that the result of \textit{T. Kusama} and \textit{J. Fujii} [ibid. 10, 133-137 (1987; Zbl 0639.62004)] does not hold if we replace test sufficiency by Bayes sufficiency.