## Annals of Statistics

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

#### Abstract

We consider situations in Bayesian analysis where we have a family of priors νh on the parameter θ, 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 θ. How do we efficiently estimate the posterior expectation of f(θ) 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 νh? 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.

#### Article information

Source
Ann. Statist., Volume 39, Number 5 (2011), 2658-2685.

Dates
First available in Project Euclid: 22 December 2011

https://projecteuclid.org/euclid.aos/1324563351

Digital Object Identifier
doi:10.1214/11-AOS913

Mathematical Reviews number (MathSciNet)
MR2906882

Zentralblatt MATH identifier
1231.62008

Subjects
Primary: 62F15: Bayesian inference 91-08: Computational methods
Secondary: 62F12: Asymptotic properties of estimators

#### Citation

Buta, Eugenia; Doss, Hani. Computational approaches for empirical Bayes methods and Bayesian sensitivity analysis. Ann. Statist. 39 (2011), no. 5, 2658--2685. doi:10.1214/11-AOS913. https://projecteuclid.org/euclid.aos/1324563351

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#### Supplemental materials

• Supplementary material: Additional technical details. We show that when estimating the Bayes factors using control variates, the estimate that is optimal when the samples are i.i.d. sequences is no longer optimal when the samples are Markov chains. We also give technical arguments regarding the consistency of spectral estimates of the variance of our estimators.