Remarks on Bayes Sufflciency
Takeru SUZUKI, Sakutarõ Yamada · Tokyo Journal of Mathematics · 1990
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), te8t suf- ficiency and Bayes sufficiency.These notions coincide with one another in the dominated case.Ramamoorthi ([6]), and Roy and Ramamoorthi ([7]), discussed Bayes sufficiency in undominated cases, but most result8 are restricted to the countably generated subfield cases.In this note we show, by examples, that PSS does not imply Bayes sufficiency in case of a continuous a priori distribution (cf.[5]), and the existence of the smallest PSS which is Bayes sufficient.This latter example shows the result by Kusama and Fujii ([4]) does not hold if we replace test sufficiency by Bayes sufficiency.By a statistical experiment we mean a triplet $(\mathscr{F}\mathscr{J}p)$ , where $\ovalbox{\tt\small REJECT}=\{P_{\theta};\theta\in\Theta\}$ is a family of probability measures on $(\mathscr{F}_{-}\ddagger K)$ .$\theta$ is referred to as the parameter space of the experiment.Let $C$ be a sigma-field of subsets of $\Theta$ which includes the sigma-field generated by the family of mappings defined by $\theta\in\Theta\rightarrow P_{\theta}(A),$ $ A\in\leftrightarrow\Psi$ For any a priori distribution $\lambda$ on $(\Theta, C)$ , let $\ovalbox{\tt\small REJECT}*\lambda$ be a probability mea8ure on $(\mathscr{F}\times\Theta, \llcorner\ovalbox{\tt\small REJECT}\times C)$ defined by $\ovalbox{\tt\small REJECT}*x(A\times C)=\int_{c}P_{\theta}(A)dx(\theta)$ .Let $\mathscr{G}$ be a sub-sigma-field of $\llcorner\ovalbox{\tt\small REJECT}$ and $\Lambda$ be a family of a priori distri- butions on $(\Theta, C)$ .DEFINITION.va is called Bayes sufficient $w$ .$r$ .$t$ .$\Lambda$ if, for any $\lambda\in\Lambda$ , $\ovalbox{\tt\small REJECT}\times\Theta$ and $\mathscr{F}\times C$ are conditionally independent given va $\times\Theta w$ .$r$ .$t$ .