Bayesian inference with nonconjugate priors

Mei-Mei Zen, Purdue University

Abstract

The issues of the behavior of posterior distributions and robust Bayes procedures are considered when nonconjugate priors are used. Four issues are addressed in this work: (i) shape properties of posterior distributions of a multivariate normal mean when nonconjugate priors are used; (ii) explicit construction of Bayesian set estimates in high dimensional problems when the posterior distribution is starunimodal; (iii) point estimation of a multivariate normal mean when t priors are used; (iv) the behavior (risk or otherwise) of systematically chosen Bayes actions when Bayesian analysis is performed with a family of priors instead of a single prior. Under (i), we take $X \sim N\sb p(\theta,\sigma\sp2 I)$, where $\sigma\sp2$ is known and let $\theta$ have a prior which is a scale mixture of multivariate normal priors with a fixed mean $\mu$ and covariance matrix $\lambda\sb\tau\sp2$I (here $\tau\sp2$ is kept fixed and the mixture is on $\lambda$). The case with known matrices $\Sigma\sb1$ and $\Sigma\sb2$ instead of $\sigma\sp2I$, $\tau\sp2I$ respectively can be reduced to the above setup. We obtain a necessary and sufficient condition for the posterior of $\theta$ to be starunimodal for a given $X$ and also a necessary and sufficient condition for the posterior of $\theta$ to be a starunimodal for all $X$. Under (ii), we explicitly construct confidence sets when the distribution of the underlying variable is starunimodal. This part of the work is completely general and can in fact be applied to Bayes and classical problems alike. We describe infinitely many starshaped confidence sets meeting the probability requirement and then derive the set with the smallest volume. We prove that in the cases where the posteriors have homothetic contours, our methods exactly reproduce the highest posterior density sets. Under (iii), the posterior mode for a multivariate normal mean is employed as a point estimate of the mean vector when t prior is used. The risk behavior of the posterior mode is explored. In particular, we prove that for any fixed m, $\tau\sp2$ and $\sigma\sp2$, the posterior mode is minimax for moderate values of p. Under (iv), we consider the problem of estimating a Binomial parameter $\theta$ and let $\theta$ have a prior $\pi$ belonging to a suitable family $\Gamma$. Five different choices of $\Gamma$ are considered. In each case, we investigate the behavior of the midpoint of the interval of Bayes estimates. We prove that unless the prior family includes unreasonable priors, the midpoint is generally an admissible procedure and for n $\le$ 4, we demonstrate that it is also usually Bayes with respect to a prior in the original family. These attractive properties of the midpoint make it a viable option in the hard technical problem of suggesting a specific action to the user.

Degree

Ph.D.

Advisors

DasGupta, Purdue University.

Subject Area

Statistics

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