Computational approaches for empirical Bayes methods and Bayesian sensitivity analysis

Details Computational approaches for empirical Bayes methods and Bayesian sensitivity analysis Computational approaches for empirical Bayes methods and Bayesian sensitivity analysis

2012-02-23

arxiv.org/abs/1202.5160v1

Annals of Statistics 2011, Vol. 39, No. 5, 2658-2685

Mathematics

Statistics Theory

Published in at http://dx.doi.org/10.1214/11-AOS913 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of

Scientific paper

10.1214/11-AOS913

We consider situations in Bayesian analysis where we have a family of priors $\nu_h$ on the parameter $\theta$, where $h$ varies continuously over a space $\mathcal{H}$, and we deal with two related problems. The first involves sensitivity analysis and is stated as follows. Suppose we fix a function $f$ of $\theta$. How do we efficiently estimate the posterior expectation of $f(\theta)$ simultaneously for all $h$ in $\mathcal{H}$? The second problem is how do we identify subsets of $\mathcal{H}$ which give rise to reasonable choices of $\nu_h$? We assume that we are able to generate Markov chain samples from the posterior for a finite number of the priors, and we develop a methodology, based on a combination of importance sampling and the use of control variates, for dealing with these two problems. The methodology applies very generally, and we show how it applies in particular to a commonly used model for variable selection in Bayesian linear regression, and give an illustration on the US crime data of Vandaele.

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