It is argued that the Calibrated Bayesian (CB) approach to statistical inference capitalizes on the strength of Bayesian and frequentist approaches to statistical inference. In the CB approach, inferences under a particular model are Bayesian, but frequentist methods are useful for model development and model checking. In this article the CB approach is outlined. Bayesian methods for missing data are then reviewed from a CB perspective. The basic theory of the Bayesian approach, and the closely related technique of multiple imputation, is described. Then applications of the Bayesian approach to normal models are described, both for monotone and nonmonotone missing data patterns. Sequential Regression Multivariate Imputation and Penalized Spline of Propensity Models are presented as two useful approaches for relaxing distributional assumptions.

Bayesian methods are increasingly applied in these days in the theory and practice of statistics. Any Bayesian inference depends on a likelihood and a prior. Ideally one would like to elicit a prior from related sources of information or past data. However, in its absence, Bayesian methods need to rely on some “objective” or “default” priors, and the resulting posterior inference can still be quite valuable.

Not surprisingly, over the years, the catalog of objective priors also has become prohibitively large, and one has to set some specific criteria for the selection of such priors. Our aim is to review some of these criteria, compare their performance, and illustrate them with some simple examples. While for very large sample sizes, it does not possibly matter what objective prior one uses, the selection of such a prior does influence inference for small or moderate samples. For regular models where asymptotic normality holds, Jeffreys’ general rule prior, the positive square root of the determinant of the Fisher information matrix, enjoys many optimality properties in the absence of nuisance parameters. In the presence of nuisance parameters, however, there are many other priors which emerge as optimal depending on the criterion selected. One new feature in this article is that a prior different from Jeffreys’ is shown to be optimal under the chi-square divergence criterion even in the absence of nuisance parameters. The latter is also invariant under one-to-one reparameterization.

Starting with the neo-Bayesian revival of the 1950s, many statisticians argued that it was inappropriate to use Bayesian methods, and in particular subjective Bayesian methods in governmental and public policy settings because of their reliance upon prior distributions. But the Bayesian framework often provides the primary way to respond to questions raised in these settings and the numbers and diversity of Bayesian applications have grown dramatically in recent years. Through a series of examples, both historical and recent, we argue that Bayesian approaches with formal and informal assessments of priors AND likelihood functions are well accepted and should become the norm in public settings. Our examples include census-taking and small area estimation, US election night forecasting, studies reported to the US Food and Drug Administration, assessing global climate change, and measuring potential declines in disability among the elderly.

Fienberg convincingly demonstrates that Bayesian models and methods represent a powerful approach to squeezing illumination from data in public policy settings. However, no school of inference is without its weaknesses, and, in the face of the ambiguities, uncertainties, and poorly posed questions of the real world, perhaps we should not expect to find a formally correct inferential strategy which can be universally applied, whatever the nature of the question: we should not expect to be able to identify a “norm” approach. An analogy is made between George Box’s “no models are right, but some are useful,” and inferential systems.

According to Hansen, Madow and Tepping [J. Amer. Statist. Assoc. 78 (1983) 776–793], “Probability sampling designs and randomization inference are widely accepted as the standard approach in sample surveys.” In this article, reasons are advanced for the wide use of this design-based approach, particularly by federal agencies and other survey organizations conducting complex large scale surveys on topics related to public policy. Impact of Bayesian methods in survey sampling is also discussed in two different directions: nonparametric calibrated Bayesian inferences from large samples and hierarchical Bayes methods for small area estimation based on parametric models.

This comment emphasizes the importance of model checking and model fitting when making inferences about finite population quantities. It also suggests the value of using unit level models when making inferences for small subpopulations, that is, “small area” analyses.

We develop and evaluate point and interval estimates for the random effects θ_i, having made observations y_i|θ_i ∼^ind N[θ_i, V_i], i = 1, …, k that follow a two-level Normal hierarchical model. Fitting this model requires assessing the Level-2 variance A ≡ Var(θ_i) to estimate shrinkages B_i ≡ V_i / (V_i + A) toward a (possibly estimated) subspace, with B_i as the target because the conditional means and variances of θ_i depend linearly on B_i, not on A. Adjustment for density maximization, ADM, can do the fitting for any smooth prior on A. Like the MLE, ADM bases inferences on two derivatives, but ADM can approximate with any Pearson family, with Beta distributions being appropriate because shrinkage factors satisfy 0 ≤ B_i ≤ 1.

Our emphasis is on frequency properties, which leads to adopting a uniform prior on A ≥ 0, which then puts Stein’s harmonic prior (SHP) on the k random effects. It is known for the “equal variances case” V₁ = ⋯ = V_k that formal Bayes procedures for this prior produce admissible minimax estimates of the random effects, and that the posterior variances are large enough to provide confidence intervals that meet their nominal coverages. Similar results are seen to hold for our approximating “ADM-SHP” procedure for equal variances and also for the unequal variances situations checked here.

For shrinkage coefficient estimation, the ADM-SHP procedure allows an alternative frequency interpretation. Writing L(A) as the likelihood of B_i with i fixed, ADM-SHP estimates B_i as B̂_i = V_i / (V_i + Â) with  ≡ argmax (A ∗ L(A)). This justifies the term “adjustment for likelihood maximization,” ALM.