Bayesian graphical modeling provides an appealing way to obtain uncertainty estimates when inferring network structures, and much recent progress has been made for Gaussian models. For more robust inferences, it is natural to consider extensions to $t$-distribution models. We argue that the classical multivariate $t$-distribution, defined using a single latent Gamma random variable to rescale a Gaussian random vector, is of little use in more highly multivariate settings, and propose other, more flexible $t$-distributions. Using an independent Gamma-divisor for each component of the random vector defines what we term the alternative $t$-distribution. The associated model allows one to extract information from highly multivariate data even when most experiments contain outliers for some of their measurements. However, the use of this alternative model comes at increased computational cost and imposes constraints on the achievable correlation structures, raising the need for a compromise between the classical and alternative models. To this end we propose the use of Dirichlet processes for adaptive clustering of the latent Gamma-scalars, each of which may then divide a group of latent Gaussian variables. The resulting Dirichlet $t$-distribution interpolates naturally between the two extreme cases of the classical and alternative $t$-distributions and combines more appealing modeling of the multivariate dependence structure with favorable computational properties.

This paper was invited by the Editor-in-Chief of Bayesian Analysis to be presented as the 2014 Best Bayesian Analysis Paper at the Twelfth World Meeting of the International Society for Bayesian Analysis (ISBA2014), held in Cancun, Mexico, on July 14–18, 2014, with invited discussions by Babak Shahbaba and François Caron.

Scale mixtures of normals have been discussed extensively in the literature as heavy-tailed alternatives to the normal distribution for robust modeling. They have been used either as error models to handle outliers or as prior distributions to provide more reasonable shrinkage of model parameters. The proposed method by Finegold and Drton goes beyond the existing literature both in terms of application (graphical models) and methodology (Dirichlet $t$) for outlier handling. While this approach can be applied to many other problems, in this discussion I will focus on its application in Bayesian modeling of high throughput biological data.

Eliciting information from experts for use in constructing prior distributions for logistic regression coefficients can be challenging. The task is especially difficult when the model contains many predictor variables, because the expert is asked to provide summary information about the probability of “success” for many subgroups of the population. Often, however, experts are confident only in their assessment of the population as a whole. This paper is about incorporating such overall information easily into a logistic regression data analysis using $g$-priors. We present a version of the $g$-prior such that the prior distribution on the overall population logistic regression probabilities of success can be set to match a beta distribution. A simple data augmentation formulation allows implementation in standard statistical software packages.

Clustering is an important and challenging statistical problem for which there is an extensive literature. Modeling approaches include mixture models and product partition models. Here we develop a product partition model and a Bayesian model selection procedure based on Bayes factors from intrinsic priors. We also find that the choice of the prior on model space is of utmost importance, almost overshadowing the other parts of the clustering problem, and we examine the behavior of the model posterior probabilities based on different model space priors. We find, somewhat surprisingly, that procedures based on the often-used uniform prior (in which all models are given the same prior probability) lead to inconsistent model selection procedures. We examine other priors, and find that the Ewens-Pitman prior and a new prior, the hierarchical uniform prior, lead to consistent model selection procedures and have other desirable properties. Lastly, we compare the procedures on a range of examples.

We consider the behavior of Bayesian procedures that perform model selection for decomposable Gaussian graphical models when the true model is in fact non-decomposable. We examine the asymptotic behavior of the posterior when models are misspecified in this way, and find that the posterior will converge to graphical structures that are minimal triangulations of the true structure. The marginal log likelihood ratio comparing different minimal triangulations is stochastically bounded, and appears to remain data dependent regardless of the sample size. The covariance matrices corresponding to the different minimal triangulations are essentially equivalent, so model averaging is of minimal benefit. Using simulated data sets and a particular high performing Bayesian method for fitting decomposable models, feature inclusion stochastic search, we illustrate that these predictions are borne out in practice. Finally, a comparison is made to penalized likelihood methods for graphical models, which make no decomposability restriction. Despite its inability to fit the true model, feature inclusion stochastic search produces models that are competitive or superior to the penalized likelihood methods, especially at higher dimensions.

Bayesian decision theory is profoundly personalistic. It prescribes the decision $d$ that minimizes the expectation of the decision-maker’s loss function $L(d,\theta )$ with respect to that person’s opinion $\pi (\theta )$. Attempts to extend this paradigm to more than one decision-maker have generally been unsuccessful, as shown in Part A of this paper. Part B of this paper explores a different decision set-up, in which Bayesians make choices knowing that later Bayesians will make decisions that matter to the earlier Bayesians. We explore conditions under which they together can be modeled as a single Bayesian. There are three reasons for doing so:

1. To understand the common structure of various examples, in some of which the reduction to a single Bayesian is possible, and in some of which it is not. In particular, it helps to deepen our understanding of the desirability of randomization to Bayesians.

2. As a possible computational simplification. When such reduction is possible, standard expected loss minimization software can be used to find optimal actions.

3. As a start toward a better understanding of social decision-making.

A common objective of fMRI (functional magnetic resonance imaging) studies is to determine subject-specific areas of increased blood oxygenation level dependent (BOLD) signal contrast in response to a stimulus or task, and hence to infer regional neuronal activity. We posit and investigate a Bayesian approach that incorporates spatial and temporal dependence and allows for the task-related change in the BOLD signal to change dynamically over the scanning session. In this way, our model accounts for potential learning effects in addition to other mechanisms of temporal drift in task-related signals. We study the properties of the model through its performance on simulated and real data sets.

We propose a multiscale model for Gaussian noised images under a Bayesian framework for both 2-dimensional (2D) and 3-dimensional (3D) images. We use a Chinese restaurant process prior to randomly generate ties among intensity values at neighboring pixels in the image. The resulting Bayesian estimator enjoys some desirable asymptotic properties for identifying precise structures in the image. The proposed Bayesian denoising procedure is completely data-driven. A conditional conjugacy property allows analytical computation of the posterior distribution without involving Markov chain Monte Carlo (MCMC) methods, making the method computationally efficient. Simulations on Shepp-Logan phantom and Lena test images confirm that our smoothing method is comparable with the best available methods for light noise and outperforms them for heavier noise both visually and numerically. The proposed method is further extended for 3D images. A simulation study shows that the proposed method is numerically better than most existing denoising approaches for 3D images. A 3D Shepp-Logan phantom image is used to demonstrate the visual and numerical performance of the proposed method, along with the computational time. MATLAB toolboxes are made available online (both 2D and 3D) to implement the proposed method and reproduce the numerical results.