The paper introduces a Bayesian estimation method for quantile regression in univariate ordinal models. Two algorithms are presented that utilize the latent variable inferential framework of Albert and Chib (1993) and the normal-exponential mixture representation of the asymmetric Laplace distribution. Estimation utilizes Markov chain Monte Carlo simulation – either Gibbs sampling together with the Metropolis–Hastings algorithm or only Gibbs sampling. The algorithms are employed in two simulation studies and implemented in the analysis of problems in economics (educational attainment) and political economy (public opinion on extending “Bush Tax” cuts). Investigations into model comparison exemplify the practical utility of quantile ordinal models.

An objective Bayesian inference is proposed for the generalized marginal random effects model $p\left(\mathbf{x}\right|\mu ,{\sigma }_{\lambda })=f({(\mathbf{x}-\mu \mathbf{1})}^T{(\mathbf{V}+{\sigma }_{\lambda }^2\mathbf{I})}^{-1}(\mathbf{x}-\mu \mathbf{1}))/\sqrt{\det (\mathbf{V}+{\sigma }_{\lambda }^2\mathbf{I})}$. The matrix $\mathbf{V}$ is assumed to be known, and the goal is to infer $\mu$ given the observations $\mathbf{x}={(x_1,\dots ,x_n)}^T$, while ${\sigma }_{\lambda }$ is a nuisance parameter. In metrology this model has been applied for the adjustment of inconsistent data $x_1,\dots ,x_n$, where the matrix $\mathbf{V}$ contains the uncertainties quoted for $x_1,\dots ,x_n$.

We show that the reference prior for grouping $\{\mu ,{\sigma }_{\lambda }\}$ is given by $\pi (\mu ,{\sigma }_{\lambda })\propto \sqrt{{\mathbf{F}}_{22}}$, where ${\mathbf{F}}_{22}$ denotes the lower right element of the Fisher information matrix $\mathbf{F}$. We give an explicit expression for the reference prior, and we also prove propriety of the resulting posterior as well as the existence of mean and variance of the marginal posterior for $\mu$. Under the additional assumption of normality, we relate the resulting reference analysis to that known for the conventional balanced random effects model in the asymptotic case when the number of repeated within-class observations for that model tends to infinity.

We investigate the frequentist properties of the proposed inference for the generalized marginal random effects model through simulations, and we also study its robustness when the underlying distributional assumptions are violated. Finally, we apply the model to the adjustment of current measurements of the Planck constant.

Doubly robust estimators are typically constructed by combining outcome regression and propensity score models to satisfy moment restrictions that ensure consistent estimation of causal quantities provided at least one of the component models is correctly specified. Standard Bayesian methods are difficult to apply because restricted moment models do not imply fully specified likelihood functions. This paper proposes a Bayesian bootstrap approach to derive approximate posterior predictive distributions that are doubly robust for estimation of causal quantities. Simulations show that the approach performs well under various sources of misspecification of the outcome regression or propensity score models. The estimator is applied in a case study of the effect of area deprivation on the incidence of child pedestrian casualties in British cities.

The use of exploratory methods is an important step in the understanding of data. When clustering functional data, most methods use traditional clustering techniques on a vector of estimated basis coefficients, assuming that the underlying signal functions live in the $L_2$-space. Bayesian methods use models which imply the belief that some observations are realizations from some signal plus noise models with identical underlying signal functions. The method we propose differs in this respect: we employ a model that does not assume that any of the signal functions are truly identical, but possibly share many of their local features, represented by coefficients in a multiresolution wavelet basis expansion. We cluster each wavelet coefficient of the signal functions using conditionally independent Dirichlet process priors, thus focusing on exact matching of local features. We then demonstrate the method using two datasets from different fields to show broad application potential.

Graphical models can be used to characterize the dependence structure for a set of random variables. In some applications, the form of dependence varies across different subgroups. This situation arises, for example, when protein activation on a certain pathway is recorded, and a subgroup of patients is characterized by a pathological disruption of that pathway. A similar situation arises when one subgroup of patients is treated with a drug that targets that same pathway. In both cases, understanding changes in the joint distribution and dependence structure across the two subgroups is key to the desired inference. Fitting a single model for the entire data could mask the differences. Separate independent analyses, on the other hand, could reduce the effective sample size and ignore the common features. In this paper, we develop a Bayesian graphical model that addresses heterogeneity and implements borrowing of strength across the two subgroups by simultaneously centering the prior towards a global network. The key feature is a hierarchical prior for graphs that borrows strength across edges, resulting in a comparison of pathways across subpopulations (differential pathways) under a unified model-based framework. We apply the proposed model to data sets from two very different studies: histone modifications from ChIP-seq experiments, and protein measurements based on tissue microarrays.

We discuss Bayesian analysis of dynamic models customized to learning and prediction with increasingly high-dimensional time series. A new framework of simultaneous graphical dynamic models allows the decoupling of analyses into those of a parallel set of univariate time series dynamic models, while flexibly modeling time-varying, cross-series dependencies and volatilities. The strategy allows for exact analysis of univariate time series models that are then coherently linked to represent the full multivariate model. Computation uses importance sampling and variational Bayes ideas, and is ideally suited to GPU-based parallelization. The analysis and its GPU-accelerated implementation is scalable with time series dimension, as we demonstrate in an analysis of a 400-dimensional financial time series.

Poisson processes are used in various applications. In their homogeneous version, the intensity process is a deterministic constant whereas it depends on time in their inhomogeneous version. To allow for an endogenous evolution of the intensity process, we consider multiplicative intensity processes. Inference methods for such processes have been developed when the trajectories are fully observed, that is to say, when both the sizes of the jumps and the jumps instants are observed. In this paper, we deal with the case of a partially observed process: we assume that the jumps sizes are non- or partially observed whereas the time events are fully observed. Moreover, we consider the case where the initial state of the process at time 0 is unknown. The inference being strongly influenced by this quantity, we propose a sensible prior distribution on the initial state, using the probabilistic properties of the process. We illustrate the performances of our methodology on a large simulation study.

Bayesian design of experiments is a methodology for incorporating prior information into the design phase of an experiment. Unfortunately, the typical Bayesian approach to designing experiments is both numerically and analytically intractable without additional assumptions or approximations. In this paper, we discuss how Gaussian processes can be used to help alleviate the numerical issues associated with Bayesian design of experiments. We provide an example based on accelerated life tests and compare our results with large-sample methods.

Many popular statistical models for complex phenomena are intractable, in the sense that the likelihood function cannot easily be evaluated. Bayesian estimation in this setting remains challenging, with a lack of computational methodology to fully exploit modern processing capabilities. In this paper we introduce novel control variates for intractable likelihoods that can dramatically reduce the Monte Carlo variance of Bayesian estimators. We prove that our control variates are well-defined and provide a positive variance reduction. Furthermore, we show how to optimise these control variates for variance reduction. The methodology is highly parallel and offers a route to exploit multi-core processing architectures that complements recent research in this direction. Indeed, our work shows that it may not be necessary to parallelise the sampling process itself in order to harness the potential of massively multi-core architectures. Simulation results presented on the Ising model, exponential random graph models and non-linear stochastic differential equation models support our theoretical findings.

Markov chains of higher order are popular models for a wide variety of applications in natural language and DNA sequence processing. However, since the number of parameters grows exponentially with the order of a Markov chain, several alternative model classes have been proposed that allow for stability and higher rate of data compression. The common notion to these models is that they cluster the possible sample paths used to predict the next state into invariance classes with identical conditional distributions assigned to the same class. The models vary in particular with respect to constraints imposed on legitime partitions of the sample paths. Here we consider the class of sparse Markov chains for which the partition is left unconstrained a priori. A recursive computation scheme based on Delaunay triangulation of the parameter space is introduced to enable fast approximation of the posterior mode partition. Comparisons with stochastic optimization, $k$-means and nearest neighbor algorithms show that our approach is both considerably faster and leads on average to a more accurate estimate of the underlying partition. We show additionally that the criterion used in the recursive steps for comparison of triangulation cell contents leads to consistent estimation of the local structure in the sparse Markov model.

When modeling geostatistical or areal data, spatial structure is commonly accommodated via a covariance function for the former and a neighborhood structure for the latter. In both cases the resulting spatial structure is a consequence of implicit spatial grouping in that observations near in space are assumed to behave similarly. It would be desirable to develop spatial methods that explicitly model the partitioning of spatial locations providing more control over resulting spatial structures and be able to better balance local and global spatial dependence. To this end, we extend product partition models to a spatial setting so that the partitioning of locations into spatially dependent clusters is explicitly modeled. We explore the resulting spatial structure and demonstrate its flexibility in accommodating many types of spatial dependencies. We illustrate the method’s utility through simulation studies and two applications. Computational techniques with additional simulations are provided in a Supplementary Material file available online.

Page and Quintana (2016) introduce the novel methodology of spatial product partition models in order to explicitly model the partitioning of spatial locations, with the aim of balancing local and global spatial dependence. Here we first discuss Gibbs-type partitions and their connection to exchangeable product partition models and their possible use as building blocks of spatial product partition models. Then, adopting the viewpoint of extreme value theory, we focus on two approaches for modeling spatial extremes, namely hierarchical modeling based on a latent stochastic process and modeling based on max-stable processes. Additional insights and interesting findings may arise by developing the approach of Page and Quintana (2016) along these lines.