Stable Decision Problems

A decision problem is characterized by a loss function V and opinion H. The pair $(V, H)$ is said to be strongly stable iff for every sequence Fₙ →_ω H, Gₙ →_ω H and Lₙ→ V, Wₙ→ V uniformly, limε ↓ 0 lim n→ ∞ ∫ Lₙ(θ, Dₙ(ε)) dFₙ(θ) - infD ∫ Lₙ(θ, D) dFₙ(θ) = 0 for every sequence Dₙ(ε) satisfying ∫ Wₙ(θ, Dₙ(ε)) dGₙ(θ) infD ∫ Wₙ(θ, D) dGₙ(θ) + ε. We show that squared error loss is unstable with any opinion if the parameter space is the real line and that any bounded loss function V(θ, D) that is continuous in θ uniformly in D is stable with any opinion H. Finally we examine the estimation or prediction case V(θ, D) = h(θ - D), where h is continuous, nondecreasing in (0, ∞) and nonincreasing in (-∞, 0) and has bounded growth. While these conditions are not enough to assure strong stability, various conditions are given that are sufficient. We believe that stability offers the beginning of a Bayesian theory of robustness.