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POSTERIOR SENSITIVITY TO THE SAMPLING DISTRIBUTION

AND THE PRIOR: MORE THAN ONE OBSERVATION

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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 Gamma^{f} and Gamma^{P} respectively. Direct and
iterative methods are described which obtain the range of
*rho*(*f,P*) over *f* \in Gamma^{f} when prior *P* is fixed, and
also the overall range over *f* \in Gamma^{f} and
*P* \in Gamma^{P}. When multiple i.i.d. observations *X*_{1},
...., *X*_{k} 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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