Stability of Bayesian decisions. (With discussion) (Q1333147)

This paper is concerned with the question of robustness of Bayesian decision rules to misspecifications of the probability measure of the expected loss function, the stability of the decision rule. Given a loss function \(L\), a decision \(\delta_ 0\), optimal in a decision space \(D\) for a probability measure \(P_ 0\), is stable if, for any sequence \(\{P_ n\}\) of probability measures weakly converging to \(P_ 0\), \[ \lim_{{\textstyle P_ n @>w>> P_ 0}} (\rho(L, P_ n, \delta_ 0) - \inf_{\delta \in D} \rho(L, P_ n, \delta)) = 0, \] where \(\rho(L, P, \delta)\) is the expected loss. The quadratic loss function, among others, exhibits the unappealing property that any decision is always unstable. The author proposes the weaker form of stability, the \(L(\cdot, \delta_ 0)\)-stability, which restricts the admissible sequences \(\{P_ n\}\) to those which satisfy \[ \lim\sup \int L(\theta, \delta_ 0) P_ n(d\theta) \leq \int L(\theta, \delta_ 0) P_ 0(d\theta), \] i.e. those for which the tail of \(L(\cdot, \delta_ 0)\) behaves uniformly. This definition bestows \(L(\cdot, \delta_ 0)\)-stability to a class of loss functions that are monotonic, continuous transformations of the distance \(\| \theta - \delta\|\), and satisfies a Markov-type inequality.