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 (2.1.1)$$ {{P}_{\theta }}\{ {{\theta }_{1}} \leqslant \theta _{1}^{{(1 - \alpha )}}(\pi ,X)\} = 1{\text{ }} - \alpha + o({{n}^{{ - r/2}}}) $$, 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, P_θ· 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.