Partial information reference priors: Derivation and interpretations

Suppose X₁,...,X_n are IID p(· θ,ψ) where (θ,ψ)∈ℝ^d is distributed according to the prior density w(·). For estimators S_n=S(X) and T_n=T(X) assumed to be consistent for some function of θ and asymptotically normal, we examine the conditional Shannon mutual information (CSMI) between Θ and T_n given Ψ and S_n, I(Θ,T_n Ψ,S_n). It is seen there are several important special cases of this CSMI. We establish asymptotic formulas for various cases and identify the resulting noninformative reference priors. As a consequence, we develop the notion of data-dependent priors and a calibration for how close an estimator is to sufficiency.