In this paper we discuss Bayesian nonconvex penalization for sparse learning problems. We explore a nonparametric formulation for latent shrinkage parameters using subordinators which are one-dimensional Lévy processes. We particularly study a family of continuous compound Poisson subordinators and a family of discrete compound Poisson subordinators. We exemplify four specific subordinators: Gamma, Poisson, negative binomial and squared Bessel subordinators. The Laplace exponents of the subordinators are Bernstein functions, so they can be used as sparsity-inducing nonconvex penalty functions. We exploit these subordinators in regression problems, yielding a hierarchical model with multiple regularization parameters. We devise ECME (Expectation/Conditional Maximization Either) algorithms to simultaneously estimate regression coefficients and regularization parameters. The empirical evaluation of simulated data shows that our approach is feasible and effective in high-dimensional data analysis.

This paper proposes a new Bayesian multiple change-point model which is based on the hidden Markov approach. The Dirichlet process hidden Markov model does not require the specification of the number of change-points a priori. Hence our model is robust to model specification in contrast to the fully parametric Bayesian model. We propose a general Markov chain Monte Carlo algorithm which only needs to sample the states around change-points. Simulations for a normal mean-shift model with known and unknown variance demonstrate advantages of our approach. Two applications, namely the coal-mining disaster data and the real United States Gross Domestic Product growth, are provided. We detect a single change-point for both the disaster data and US GDP growth. All the change-point locations and posterior inferences of the two applications are in line with existing methods.

In this article we describe Bayesian nonparametric procedures for two-sample hypothesis testing. Namely, given two sets of samples ${\mathbf{y}}^{(1)}\stackrel{\mathrm{iid}}{~}{F}^{(1)}$ and ${\mathbf{y}}^{(2)}\stackrel{\mathrm{iid}}{~}{F}^{(2)}$, with $F^{(1)},F^{(2)}$ unknown, we wish to evaluate the evidence for the null hypothesis $H_0:F^{(1)}\equiv F^{(2)}$ versus the alternative $H_1:F^{(1)}\ne F^{(2)}$. Our method is based upon a nonparametric Pólya tree prior centered either subjectively or using an empirical procedure. We show that the Pólya tree prior leads to an analytic expression for the marginal likelihood under the two hypotheses and hence an explicit measure of the probability of the null $\mathrm{Pr}(H_0|\{{\mathbf{y}}^{(1)},{\mathbf{y}}^{(2)}\}\mathbf{)}$

Prior sensitivity examination plays an important role in applied Bayesian analyses. This is especially true for Bayesian hierarchical models, where interpretability of the parameters within deeper layers in the hierarchy becomes challenging. In addition, lack of information together with identifiability issues may imply that the prior distributions for such models have an undesired influence on the posterior inference. Despite its importance, informal approaches to prior sensitivity analysis are currently used. They require repetitive re-fits of the model with ad-hoc modified base prior parameter values. Other formal approaches to prior sensitivity analysis suffer from a lack of popularity in practice, mainly due to their high computational cost and absence of software implementation. We propose a novel formal approach to prior sensitivity analysis, which is fast and accurate. It quantifies sensitivity without the need for a model re-fit. Through a series of examples we show how our approach can be used to detect high prior sensitivities of some parameters as well as identifiability issues in possibly over-parametrized Bayesian hierarchical models.

Gaussian concentration graph models and covariance graph models are two classes of graphical models that are useful for uncovering latent dependence structures among multivariate variables. In the Bayesian literature, graphs are often determined through the use of priors over the space of positive definite matrices with fixed zeros, but these methods present daunting computational burdens in large problems. Motivated by the superior computational efficiency of continuous shrinkage priors for regression analysis, we propose a new framework for structure learning that is based on continuous spike and slab priors and uses latent variables to identify graphs. We discuss model specification, computation, and inference for both concentration and covariance graph models. The new approach produces reliable estimates of graphs and efficiently handles problems with hundreds of variables.

We consider a study of players employed by teams who are members of the National Basketball Association where units of observation are functional curves that are realizations of production measurements taken through the course of one’s career. The observed functional output displays large amounts of between player heterogeneity in the sense that some individuals produce curves that are fairly smooth while others are (much) more erratic. We argue that this variability in curve shape is a feature that can be exploited to guide decision making, learn about processes under study and improve prediction. In this paper we develop a methodology that takes advantage of this feature when clustering functional curves. Individual curves are flexibly modeled using Bayesian penalized B-splines while a hierarchical structure allows the clustering to be guided by the smoothness of individual curves. In a sense, the hierarchical structure balances the desire to fit individual curves well while still producing meaningful clusters that are used to guide prediction. We seamlessly incorporate available covariate information to guide the clustering of curves non-parametrically through the use of a product partition model prior for a random partition of individuals. Clustering based on curve smoothness and subject-specific covariate information is particularly important in carrying out the two types of predictions that are of interest, those that complete a partially observed curve from an active player, and those that predict the entire career curve for a player yet to play in the National Basketball Association.

Approximate Bayesian Computation (ABC) is a useful class of methods for Bayesian inference when the likelihood function is computationally intractable. In practice, the basic ABC algorithm may be inefficient in the presence of discrepancy between prior and posterior. Therefore, more elaborate methods, such as ABC with the Markov chain Monte Carlo algorithm (ABC-MCMC), should be used. However, the elaboration of a proposal density for MCMC is a sensitive issue and very difficult in the ABC setting, where the likelihood is intractable. We discuss an automatic proposal distribution useful for ABC-MCMC algorithms. This proposal is inspired by the theory of quasi-likelihood (QL) functions and is obtained by modelling the distribution of the summary statistics as a function of the parameters. Essentially, given a real-valued vector of summary statistics, we reparametrize the model by means of a regression function of the statistics on parameters, obtained by sampling from the original model in a pilot-run simulation study. The QL theory is well established for a scalar parameter, and it is shown that when the conditional variance of the summary statistic is assumed constant, the QL has a closed-form normal density. This idea of constructing proposal distributions is extended to non constant variance and to real-valued parameter vectors. The method is illustrated by several examples and by an application to a real problem in population genetics.

A Multiregression Dynamic Model (MDM) is a class of multivariate time series that represents various dynamic causal processes in a graphical way. One of the advantages of this class is that, in contrast to many other Dynamic Bayesian Networks, the hypothesised relationships accommodate conditional conjugate inference. We demonstrate for the first time how straightforward it is to search over all possible connectivity networks with dynamically changing intensity of transmission to find the Maximum a Posteriori Probability (MAP) model within this class. This search method is made feasible by using a novel application of an Integer Programming algorithm. The efficacy of applying this particular class of dynamic models to this domain is shown and more specifically the computational efficiency of a corresponding search of 11-node Directed Acyclic Graph (DAG) model space. We proceed to show how diagnostic methods, analogous to those defined for static Bayesian Networks, can be used to suggest embellishment of the model class to extend the process of model selection. All methods are illustrated using simulated and real resting-state functional Magnetic Resonance Imaging (fMRI) data.

Bayesian model selection with improper priors is not well-defined because of the dependence of the marginal likelihood on the arbitrary scaling constants of the within-model prior densities. We show how this problem can be evaded by replacing marginal log-likelihood by a homogeneous proper scoring rule, which is insensitive to the scaling constants. Suitably applied, this will typically enable consistent selection of the true model.

This note is a discussion of the article “Bayesian model selection based on proper scoring rules” by A. P. Dawid and M. Musio, to appear in Bayesian Analysis. While appreciating the concepts behind the use of proper scoring rules, we point out here some possible practical difficulties with the advocated approach.

We are deeply appreciative of the initiative of the editor, Marina Vanucci, in commissioning a discussion of our paper, and extremely grateful to all the discussants for their insightful and thought-provoking comments. We respond to the discussions in alphabetical order.