POSTERIOR SENSITIVITY TO THE SAMPLING DISTRIBUTION AND THE PRIOR: MORE THAN ONE OBSERVATION

SANJIB BASU

Department of Mathematical Sciences, University of Arkansas, Fayetteville, AR 72701, U.S.A.

(Received April 28, 1995; revised March 13, 1998)

Abstract. Sensitivity of a posterior quantity ρ(f, P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of ρ(f, P) when f and P vary in nonparametric classes Γ^{f} and Γ^{P} respectively. Direct and iterative methods are described which obtain the range of ρ(f, P) over f ∈ Γ^{f} when prior P is fixed, and also the overall range over f ∈ Γ^{f} and P ∈ Γ^{P}. When multiple i.i.d. observations X_{1}, …, X_{k} are observed from f, the posterior quantity ρ(f, P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of ρ(f, P). Several examples illustrate applications of these methods.

Key words and phrases: Bayesian robustness, Gâteaux derivative, model robustness, model selection, predictive distribution, prior robustness, steepest descent.