Bayesian statistical practice makes extensive use of versions of objective Bayesian analysis. We discuss why this is so, and address some of the criticisms that have been raised concerning objective Bayesian analysis. The dangers of treating the issue too casually are also considered. In particular, we suggest that the statistical community should accept formal objective Bayesian techniques with confidence, but should be more cautious about casual objective Bayesian techniques.

We address the position of subjectivism within Bayesian statistics. We argue, first, that the subjectivist Bayes approach is the only feasible method for tackling many important practical problems. Second, we describe the essential role of the subjectivist approach in scientific analysis. Third, we consider possible modifications to the Bayesian approach from a subjectivist viewpoint. Finally, we address the issue of pragmatism in implementing the subjectivist approach.

In this contribution to the discussion of "The case for objective Bayesian analysis" by James Berger and "Subjective Bayesian analysis: principles and practice" by Michael Goldstein, I argue that (a) all Bayesian work is inherently subjective and needs to be guided simultaneously by considerations of both coherence and calibration, and (b) "objective" (diffuse) prior distributions are sometimes, but not always, useful in attaining good calibrative performance---it depends (as usual) on your judgment about how knowns (e.g., past observables) and unknowns (e.g., future observables) are related.

The subjective-objective dialogue between Goldstein (2006) and Berger (2006) lays out strong cases for what seem to be two schools of Bayesian thought. But a closer look suggests to me that while both authors address the pragmatics of their approaches, only one qualifies as a school of thought. In these comments I address briefly seven dimensions: the history of Bayesian thought, the different roles for a Bayesian approach, the subjectivity of scientists and the illusion of objectivity, the subjectivity of the likelihood function, the difficulty in separating likelihood from prior, pragmatism, and the fruitless search for the objective prior.

The dangerous heresy of so-called 'objective' Bayesian methods is again propounded by Berger. These comments are my attempt to save Bayesian statistics.

I have deliberately chosen rather dramatic, perhaps inflammatory, language in the above sentences. I do not expect all readers to view Berger's proposals in the same terms, but I hope that they will find my comments thought-provoking and constructive. It is undoubtedly true that the use of weakly informative prior distributions is both essential and valuable in practice. However, it is vitally important that their role is properly understood, instead of being grossly overstated.

My comments continue with some thoughts about Bayesian software that I hope are in tune with the agenda of the 'objective Bayesians', and certainly with the goal of spreading Bayesian methods more widely.

In this comment, I argue that Bayes procedures with good frequentist properties are objective. I introduce the idea with a short play, followed by some commentary.

We use simulation studies, whose design is realistic for educational and medical research (as well as other fields of inquiry), to compare Bayesian and likelihood-based methods for fitting variance-components (VC) and random-effects logistic regression (RELR) models. The likelihood (and approximate likelihood) approaches we examine are based on the methods most widely used in current applied multilevel (hierarchical) analyses: maximum likelihood (ML) and restricted ML (REML) for Gaussian outcomes, and marginal and penalized quasi-likelihood (MQL and PQL) for Bernoulli outcomes. Our Bayesian methods use Markov chain Monte Carlo (MCMC) estimation, with adaptive hybrid Metropolis-Gibbs sampling for RELR models, and several diffuse prior distributions ($\Gamma^{ -1 }( \epsilon, \epsilon )$ and $U( 0, \frac{ 1 }{ \epsilon } )$ priors for variance components). For evaluation criteria we consider bias of point estimates and nominal versus actual coverage of interval estimates in repeated sampling. In two-level VC models we find that (a) both likelihood-based and Bayesian approaches can be made to produce approximately unbiased estimates, although the automatic manner in which REML accomplishes this is an advantage, but (b) both approaches had difficulty achieving nominal coverage in small samples and with small values of the intraclass correlation. With the three-level RELR models we examine we find that (c) quasi-likelihood methods for estimating random-effects variances perform badly with respect to bias and coverage in the example we simulated, and (d) Bayesian diffuse-prior methods lead to well-calibrated point and interval RELR estimates. While it is true that the likelihood-based methods we study are considerably faster computationally than MCMC, (i) steady improvements in recent years in both hardware speed and efficiency of Monte Carlo algorithms and (ii) the lack of calibration of likelihood-based methods in some common hierarchical settings combine to make MCMC-based Bayesian fitting of multilevel models an attractive approach, even with rather large data sets. Other analytic strategies based on less approximate likelihood methods are also possible but would benefit from further study of the type summarized here.

Various noninformative prior distributions have been suggested for scale parameters in hierarchical models. We construct a new folded-noncentral-$t$ family of conditionally conjugate priors for hierarchical standard deviation parameters, and then consider noninformative and weakly informative priors in this family. We use an example to illustrate serious problems with the inverse-gamma family of "noninformative" prior distributions. We suggest instead to use a uniform prior on the hierarchical standard deviation, using the half-$t$ family when the number of groups is small and in other settings where a weakly informative prior is desired. We also illustrate the use of the half-$t$ family for hierarchical modeling of multiple variance parameters such as arise in the analysis of variance.

For a scalar random-effect variance, Browne and Draper (2005) have found that the uniform prior works well. It would be valuable to know more about the vector case, in which a second-stage prior on the random effects variance matrix ${\bf D}$ is needed. We suggest consideration of an inverse Wishart prior for ${\bf D}$ where the scale matrix is determined from the first-stage variance.

The power prior has emerged as a useful informative prior for the incorporation of historical data in a Bayesian analysis. Viewing hierarchical modeling as the "gold standard" for combining information across studies, we provide a formal justification of the power prior by examining formal analytical relationships between the power prior and hierarchical modeling in linear models. Asymptotic relationships between the power prior and hierarchical modeling are obtained for non-normal models, including generalized linear models, for example. These analytical relationships unify the theory of the power prior, demonstrate the generality of the power prior, shed new light on benchmark analyses, and provide insights into the elicitation of the power parameter in the power prior. Several theorems are presented establishing these formal connections, as well as a formal methodology for eliciting a guide value for the power parameter $a_0$ via hierarchical models.

We propose a Bayesian semiparametric accelerated failure time (AFT) model in which the baseline survival distribution is modeled as a Dirichlet process mixture of gamma densities. The model is highly flexible and readily captures features such as multimodality in predictive survival densities. The approach can be used in a "black-box" manner in that the prior information needed to fit the model can be quite vague, and we recommend a particular prior in the absence of information on the baseline survival distribution. The resulting posterior baseline distribution has mass only on the positive reals, a desirable feature in a failure-time model. The formulae needed to fit the model are available in closed-form and the model is relatively easy to code and implement. We provide both simulated and real data examples, including data on the cosmetic effects of cancer therapy.

Spatial cumulative distributions (SCDFs) are useful in environmental applications -- for example, by helping assess the fraction of a region exposed to harmful pollutants. Data sets containing the requisite spatial information often contain temporal data as well. We therefore extend the notion of an SCDF to a spatiotemporal cumulative distribution function (STCDF), with the goal of increasing precision by making use of repeated measurements. Ours is a hierarchical Bayesian approach, with estimation carried out by Markov chain Monte Carlo (MCMC) methods. We develop linear algebra results and corresponding computational techniques to handle the difficulties in evaluating the likelihood wrought by the large data sets (due to the added temporal component), the inclusion of spatial and temporal random effects, the need to account for measurement error, and the handling of missing data. We illustrate the concepts in a univariate setting with an Atlanta ozone data set, and in a bivariate (two pollutant) setting with a California NO/NO$_2$ data set.

In this paper we propose a generalised iterative algorithm for calculating variational Bayesian estimates for a normal mixture model and investigate its convergence properties. It is shown theoretically that the variational Bayesian estimator converges locally to the maximum likelihood estimator at the rate of $O(1/{n})$ in the large sample limit.