Consistency of posterior mixtures in the Gaussian family on a Hilbert space and its applications (Q1907817)

In ibid. 48, No. 1, 87-106 (1994; Zbl 0797.62005), we considered the compound estimation problem of \textit{H. Robbins} [Proc. Berkeley Symp. Math. Stat. Probab., California 1950, 131-148 (1951; Zbl 0044.14803)] in the mean-parameter family of Gaussian distributions on a real separable infinite dimensional Hilbert space \(H\), the common covariance operator \(C\) of the distributions is assumed to be known. Since a Gaussian distribution with mean \(\theta\) is either mutually absolutely continuous (equivalent) or singular to the Gaussian distribution with mean \(0\), with equivalence iff \(\theta\in H_0= C^{1/2}H\), for the purpose of inference it suffices to restrict attention to the subfamily for which the mean is in \(H_0\). In the following Section 2, we summarize certain necessary facts about Gaussian distributions on a real separable infinite dimensional Hilbert space from our earlier paper. In Section 3, a consistency result is developed; it is proved in Section 4. In Section 5, an application of the consistency result to the empirical Bayes problem is discussed.