Some Orthogonality Preserving Kernels which are not Completely Orthogonal

Abstract

It is shown that a perturbed Bernoulli probability transition kernel yields an explicit example of an orthogonality preserving kernel which is not completely orthogonal. In statistical language, such a kernel defines models $P_\theta, \theta \in \lbrack 0, 1 \rbrack$, in which there is no estimate that estimates $\theta$ perfectly for all $\theta$, but there is, for any given prior distribution on $\theta$ and hypothesis $H_0 \subset \lbrack 0, 1 \rbrack$, a perfect test for $H_0$ against its complement $\lbrack 0, 1\rbrack\backslash H_0$. It is also demonstrated with an analysis and an application of sets and maps with the Baire property that there are continuum many nonisomorphic atomless orthogonality preserving transition kernels which are not completely orthogonal. Our methods may be regarded as refinements of those used by Blackwell.