### 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 rho(f,P) to the choice of the sampling distribution f and prior P is considered. Sensitivity is measured by the range of rho(f,P) when f and P vary in nonparametric classes Gammaf and GammaP respectively. Direct and iterative methods are described which obtain the range of rho(f,P) over f \in Gammaf when prior P is fixed, and also the overall range over f \in Gammaf and P \in GammaP. When multiple i.i.d. observations X1, ...., Xk are observed from f, the posterior quantity rho(f,P) is not a ratio-linear function of f. A method of steepest descent is proposed to obtain the range of rho(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.

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