On the existence of inferences which are consistent with a given model

Abstract

4 If ${p_{\theta}$ is a $\sigma$-additive statistical model and $\pi$ a finitely additive prior, then any statistic T is sufficient, with respect to a suitable inference consistent with ${p_{\theta}$ and $\pi$, provided only that $p_{\theta}(T = t) = 0$ for all $\theta$ and t. Here, sufficiency is to be intended in the Bayesian sense, and consistency in the sense of Lane and Sudderth. As a corollary, if ${p_{\theta}$ is $\sigma$-additive and diffuse, then, whatever the prior $\pi$, there is an inference which is consistent with ${p_{\theta}$ and $\pi$. Two versions of the main result are also obtained for predictive problems.