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Probability matching priors, ensuring frequentist validity of posterior credible sets up to the desired order of asymptotics, are of substantial current interest. They can form the basis of an objective Bayesian analysis. In addition, they provide a route for obtaining accurate frequentist confidence sets, which are meaningful also to a Bayesian. This monograph presents, for the first time in book form, an up-to-date and comprehensive account of probability matching priors addressing the problems of both estimation and prediction. Apart from being useful to researchers, it can be the core of a one-semester graduate course in Bayesian asymptotics.

Gauri Sankar Datta is a professor of statistics at the University of Georgia. He has published extensively in the fields of Bayesian analysis, likelihood inference, survey sampling, and multivariate analysis.

Rahul Mukerjee is a professor of statistics at the Indian Institute of Management Calcutta. He co-authored three other research monographs, including "A Calculus for Factorial Arrangements" in this series. A fellow of the Institute of Mathematical Statistics, Dr. Mukerjee is on the editorial boards of several international journals.



1. Introduction and the Shrinkage Argument

The study of priors ensuring, up to the desired order of asymptotics, the approximate frequentist validity of posterior credible sets has received significant attention in recent years and a considerable interest is still continuing in this field. Bayesian credible sets based on these priors have approximately correct frequentist coverage as well. Such priors are generically known as probability matching priors, or matching priors in short. As noted by Tibshirani (1989) among others, study in this direction has several important practical implications with appeal to both Bayesians and frequentists:
First, the ensuing matching priors are, in a sense, noninformative. The approximate agreement between the Bayesian and frequentist coverage probabilities of the associated credible sets provides an external validation for these priors. They can form the basis of an objective Bayesian analysis and are potentially useful for comparative purposes in subjective Bayesian analyses as well.
Second, Bayesian credible sets given by matching priors can also be interpreted as accurate frequentist confidence sets because of their approximately correct frequentist coverage. Thus the exploration of matching priors provides a route for obtaining accurate frequentist confidence sets which are meaningful also to a Bayesian.
In addition, research in this area has led to the development of a powerful and transparent Bayesian route, via a shrinkage argument, for higher order asymptotic frequentist computations.
Gauri Sankar Datta, Rahul Mukerjee

2. Matching Priors for Posterior Quantiles

The early literature on matching priors centered around those which ensure approximate frequentist validity of the posterior quantiles of a onedimensional interest parameter (Welch and Peers, 1963; Peers, 1965). Even a major part of the recent research on matching priors has been on priors of this kind. In the present chapter, we review these developments. Specifically, we shall be considering priors π(·) for which the relation
$$ {{P}_{\theta }}\{ {{\theta }_{1}} \leqslant \theta _{1}^{{(1 - \alpha )}}(\pi ,X)\} = 1{\text{ }} - \alpha + o({{n}^{{ - r/2}}}) $$
, holds for r = 1 or 2 and for each α (0 < α < 1). Here n is the sample size, \(\theta = (\theta _1,\cdots,\theta_p)^T\) is an unknown parameter vector, θ1 is the one-dimensional parameter of interest, Pθ· is the frequentist probability measure under θ, and \(\theta _1^{(1-\alpha)} (\pi, X)\) is the (1-α)th posterior quantile of θ1, under π(·), given the data X. Of course, we require (2.1.1) and its counterparts considered later in this chapter to hold for all possible θ as well, a point which is implicit throughout. Priors satisfying (2.1.1) for r = 1 or 2 are called first or second order matching priors, respectively. Clearly, they ensure that one-sided Bayesian credible sets of the form \((- \infty, \theta_1^{(1-\alpha)} (\pi, X)]\) for θ1 have correct frequentist coverage as well, up to the order of approximation indicated in (2.1.1). As will be seen later, for p ≥ 2, i.e., in the presence of nuisance parameters, a first order matching prior is not unique. The study of second order matching priors, which ensure correct frequentist coverage to a higher order of approximation, can help in significantly narrowing down the class of competing first order matching priors.
Gauri Sankar Datta, Rahul Mukerjee

3. Matching Priors for Distribution Functions

Matching priors for posterior quantiles were discussed at length in the previous chapter. These priors concern a single parameter (see Theorem 2.4.1) or a single parametric function (see Theorem 2.8.1) of interest. Since quantiles are intimately linked with the cumulative distribution function (c.d.f.), one may wonder how far these results carry through when matching is done via c.d.f. instead of quantiles. Continuing with a one-dimensional parameter or parametric function of interest, this issue is addressed in Section 3.2. The results in this section lead to the satisfying conclusion that first order matching priors for quantiles remain so when the analysis is based on a comparison of the posterior and frequentist c.d.f.’s.
Gauri Sankar Datta, Rahul Mukerjee

4. Matching Priors for Highest Posterior Density Regions

Highest posterior density (HPD) regions are very popular with Bayesians. With a possibly multidimensional interest parameter θ, such a region is of the form
$$\{\tilde\theta: \pi (\tilde{\theta}|X) \geq K\}$$
, where \(\pi (\tilde{\theta}|X)\) is the posterior density of θ, under a prior π(·), given the data X, and K depends on π(·) and X in addition to the chosen posterior credibility level. Clearly, by the Neyman-Pearson lemma, an HPD region has the smallest possible volume, given X, at a chosen level of credibility. In this chapter, we consider priors ensuring approximate frequentist validity of HPD regions with margin of error o(n -1 ), where n is the sample size. Priors of this kind are called matching priors for HPD regions or, briefly, HPD matching priors. They can be useful even when the interest parameter is multidimensional since HPD regions are well-defined in such situations.
Gauri Sankar Datta, Rahul Mukerjee

5. Matching Priors for Other Credible Regions

In this chapter, we focus on posterior credible regions obtained by the inversion of certain commonly used statistics. Priors ensuring approximate frequentist validity of such regions are characterized. This is done with margin of error o(n-1), where n is the sample size. The results, when combined with those of the previous chapter, can help in narrowing down the choice of matching priors especially when the interest parameter is multidimensional.
Gauri Sankar Datta, Rahul Mukerjee

6. Matching Priors for Prediction

In the preceding chapters, we considered probability matching priors for estimation. The object of interest was a parameter, either one-dimensional or multidimensional, and priors ensuring approximate frequentist validity of the associated posterior credible regions were studied. Evidently, the solutions for these matching priors depend on the specification of the interest parameter. For instance, in Example 2.5.7 concerning the Student’s t-model, it was seen that a unique second order matching prior exists when the location parameter θ1 is of interest whereas no such prior is available when interest lies in the shape parameter θ2.
Gauri Sankar Datta, Rahul Mukerjee


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