Showing papers in "Bayesian Analysis in 2012"

On the half-cauchy prior for a global scale parameter

Abstract: This paper argues that the half-Cauchy distribution should replace the inverseGamma distribution as a default prior for a top-level scale parameter in Bayesian hierarchical models, at least for cases where a proper prior is necessary. Our arguments involve a blend of Bayesian and frequentist reasoning, and are intended to complement the original case made by Gelman (2006) in support of the folded-t family of priors. First, we generalize the half-Cauchy prior to the wider class of hypergeometric inverted-beta priors. We derive expressions for posterior moments and marginal densities when these priors are used for a top-level normal variance in a Bayesian hierarchical model. We go on to prove a proposition that, together with the results for moments and marginals, allows us to characterize the frequentist risk of the Bayes estimators under all global-shrinkage priors in the class. These theoretical results, in turn, allow us to study the frequentist properties of the half-Cauchy prior versus a wide class of alternatives. The half-Cauchy occupies a sensible “middle ground” within this class: it performs very well near the origin, but does not lead to drastic compromises in other parts of the parameter space. This provides an alternative, classical justification for the repeated, routine use of this prior. We also consider situations where the underlying mean vector is sparse, where we argue that the usual conjugate choice of an inverse-gamma prior is particularly inappropriate, and can lead to highly distorted posterior inferences. Finally, we briefly summarize some open issues in the specification of default priors for scale terms in hierarchical models.

Scalable Variational Inference for Bayesian Variable Selection in Regression, and Its Accuracy in Genetic Association Studies

Abstract: The Bayesian approach to variable selection in regression is a powerful tool for tackling many scientific problems. Inference for variable selection models is usually implemented using Markov chain Monte Carlo (MCMC). Because MCMC can impose a high computational cost in studies with a large number of variables, we assess an alternative to MCMC based on a simple variational approximation. Our aim is to retain useful features of Bayesian variable selection at a reduced cost. Using simulations designed to mimic genetic association studies, we show that this simple variational approximation yields posterior inferences in some settings that closely match exact values. In less restrictive (and more realistic) conditions, we show that posterior probabilities of inclusion for individual variables are often incorrect, but variational estimates of other useful quantities|including posterior distributions of the hyperparameters|are remarkably accurate. We illustrate how these results guide the use of variational inference for a genome-wide association study with thousands of samples and hundreds of thousands of variables.

Bayesian Graphical Lasso Models and Efficient Posterior Computation

Abstract: Recently, the graphical lasso procedure has become popular in estimating Gaussian graphical models. In this paper, we introduce a fully Bayesian treatment of graphical lasso models. We first investigate the graphical lasso prior that has been relatively unexplored. Using data augmentation, we develop a simple but highly efficient block Gibbs sampler for simulating covariance matrices. We then generalize the Bayesian graphical lasso to the Bayesian adaptive graphical lasso. Finally, we illustrate and compare the results from our approach to those obtained using the standard graphical lasso procedures for real and simulated data. In terms of both covariance matrix estimation and graphical structure learning, the Bayesian adaptive graphical lasso appears to be the top overall performer among a range of frequentist and Bayesian methods.

Commensurate Priors for Incorporating Historical Information in Clinical Trials Using General and Generalized Linear Models.

Abstract: Assessing between-study variability in the context of conventional random-effects meta-analysis is notoriously difficult when incorporating data from only a small number of historical studies In order to borrow strength, historical and current data are often assumed to be fully homogeneous, but this can have drastic consequences for power and Type I error if the historical information is biased In this paper, we propose empirical and fully Bayesian modifications of the commensurate prior model (Hobbs et al, 2011) extending Pocock (1976), and evaluate their frequentist and Bayesian properties for incorporating patient-level historical data using general and generalized linear mixed regression models Our proposed commensurate prior models lead to preposterior admissible estimators that facilitate alternative bias-variance trade-offs than those offered by pre-existing methodologies for incorporating historical data from a small number of historical studies We also provide a sample analysis of a colon cancer trial comparing time-to-disease progression using a Weibull regression model

Spatial Quantile Multiple Regression Using the Asymmetric Laplace Process

Abstract: We consider quantile multiple regression through conditional quantile models, i.e. each quantile is modeled separately. We work in the context of spatially referenced data and extend the asymmetric Laplace model for quantile regression to a spatial process, the asymmetric Laplace process (ALP) for quantile regression with spatially dependent errors. By taking advantage of a convenient conditionally Gaussian representation of the asymmetric Laplace distribution, we are able to straightforwardly incorporate spatial dependence in this process. We develop the properties of this process under several specifications, each of which induces different smoothness and covariance behavior at the extreme quantiles. We demonstrate the advantages that may be gained by incorporating spatial dependence into this conditional quantile model by applying it to a data set of log selling prices of homes in Baton Rouge, LA, given characteristics of each house. We also introduce the asymmetric Laplace predictive process (ALPP) which accommodates large data sets, and apply it to a data set of birth weights given maternal covariates for several thousand births in North Carolina in 2000. By modeling the spatial structure in the data, we are able to show, using a check loss function, improved performance on each of the data sets for each of the quantiles at which the model was fit.

Combining Expert Opinions in Prior Elicitation

Abstract: We consider the problem of combining opinions from different experts in an explicitly model-based way to construct a valid subjective prior in a Bayesian statistical approach. We propose a generic approach by considering a hierarchical model accounting for various sources of variation as well as accounting for potential dependence between experts. We apply this approach to two problems. The first problem deals with a food risk assessment problem involving modelling dose-response for Listeria monocytogenes contamination of mice. Two hierarchical levels of variation are considered (between and within experts) with a complex mathematical situation due to the use of an indirect probit regression. The second concerns the time taken by PhD students to submit their thesis in a particular school. It illustrates a complex situation where three hierarchical levels of variation are modelled but with a simpler underlying probability distribution (log-Normal).

A New Robust Regression Model for Proportions

Abstract: A new regression model for proportions is presented by considering the Beta rectangular distribution proposed by Hahn (2008). This new model includes the Beta regression model introduced by Ferrari and Cribari-Neto (2004) and the variable dispersion Beta regression model introduced by Smithson and Verkuilen (2006) as particular cases. Like Branscum, Johnson, and Thurmond (2007), a Bayesian inference approach is adopted using Markov Chain Monte Carlo (MCMC) algorithms. Simulation studies on the influence of outliers by considering contaminated data under four perturbation patterns to generate outliers were carried out and confirm that the Beta rectangular regression model seems to be a new robust alternative for modeling proportion data and that the Beta regression model shows sensitivity to the estimation of regression coefficients, to the posterior distribution of all parameters and to the model comparison criteria considered. Furthermore, two applications are presented to illustrate the robustness of the Beta rectangular model.

Efficient MCMC for Climate Model Parameter Estimation: Parallel Adaptive Chains and Early Rejection

Abstract: The emergence of Markov chain Monte Carlo (MCMC) methods has opened a way for Bayesian analysis of complex models. Running MCMC samplers typically requires thousands of model evaluations, which can exceed available computer resources when this evaluation is computationally intensive. We will discuss two generally applicable techniques to improve the efficiency of MCMC. First, we consider a parallel version of the adaptive MCMC algorithm of Haario et al. (2001), implementing the idea of inter-chain adaptation introduced by Craiu et al. (2009). Second, we present an early rejection (ER) approach, where model simulation is stopped as soon as one can conclude that the proposed parameter value will be rejected by the MCMC algorithm. This work is motivated by practical needs in estimating parameters of climate and Earth system models. These computationally intensive models involve non-linear expressions of the geophysical and biogeochemical processes of the Earth system. Modeling of these processes, especially those operating in scales smaller than the model grid, involves a number of specified parameters, or ‘tunables’. MCMC methods are applicable for estimation of these parameters, but they are computationally very demanding. Efficient MCMC variants are thus needed to obtain reliable results in reasonable time. Here we evaluate the computational gains attainable through parallel adaptive MCMC and Early Rejection using both simple examples and a realistic climate model.

Beta Processes, Stick-Breaking and Power Laws

Abstract: The beta-Bernoulli process provides a Bayesian nonparametric prior for models involving collections of binary-valued features. A draw from the beta process yields an infinite collection of probabilities in the unit interval, and a draw from the Bernoulli process turns these into binary-valued features. Recent work has provided stick-breaking representations for the beta process analogous to the well-known stick-breaking representation for the Dirichlet process. We derive one such stick-breaking representation directly from the characterization of the beta process as a completely random measure. This approach motivates a three-parameter generalization of the beta process, and we study the power laws that can be obtained from this generalized beta process. We present a posterior inference algorithm for the beta-Bernoulli process that exploits the stick-breaking representation, and we present experimental results for a discrete factor-analysis model.

Simultaneous Linear Quantile Regression: A Semiparametric Bayesian Approach

Abstract: We introduce a semi-parametric Bayesian framework for a simultaneous analysis of linear quantile regression models. A simultaneous analysis is essential to attain the true potential of the quantile regression framework, but is computa- tionally challenging due to the associated monotonicity constraint on the quantile curves. For a univariate covariate, we present a simpler equivalent characterization of the monotonicity constraint through an interpolation of two monotone curves. The resulting formulation leads to a tractable likelihood function and is embedded within a Bayesian framework where the two monotone curves are modeled via lo- gistic transformations of a smooth Gaussian process. A multivariate extension is suggested by combining the full support univariate model with a linear projection of the predictors. The resulting single-index model remains easy to flt and provides substantial and measurable improvement over the flrst order linear heteroscedastic model. Two illustrative applications of the proposed method are provided.

On the Support of MacEachern’s Dependent Dirichlet Processes and Extensions

Abstract: We study the support properties of Dirichlet process–based models for sets of predictor–dependent probability distributions. Exploiting the connection between copulas and stochastic processes, we provide an alternative definition of MacEachern’s dependent Dirichlet processes. Based on this definition, we provide sufficient conditions for the full weak support of different versions of the process. In particular, we show that under mild conditions on the copula functions, the version where only the support points or the weights are dependent on predictors have full weak support. In addition, we also characterize the Hellinger and Kullback–Leibler support of mixtures induced by the different versions of the dependent Dirichlet process. A generalization of the results for the general class of dependent stick–breaking processes is also provided.

Simulation-based Regularized Logistic Regression

Abstract: In this paper, we develop a simulation-based framework for regu- larized logistic regression, exploiting two novel results for scale mixtures of nor- mals. By carefully choosing a hierarchical model for the likelihood by one type of mixture, and implementing regularization with another, we obtain new MCMC schemes with varying e-ciency depending on the data type (binary v. binomial, say) and the desired estimator (maximum likelihood, maximum a posteriori, poste- rior mean). Advantages of our omnibus approach include ∞exibility, computational e-ciency, applicability in p ? n settings, uncertainty estimates, variable selection, and assessing the optimal degree of regularization. We compare our methodology to modern alternatives on both synthetic and real data. An R package called reglogit is available on CRAN.

Mixture Modeling for Marked Poisson Processes

Abstract: We propose a general inference framework for marked Poisson processes observed over time or space. Our modeling approach exploits the connection of nonhomogeneous Poisson process intensity with a density function. Nonparametric Dirichlet process mixtures for this density, combined with nonparametric or semiparametric modeling for the mark distribution, yield flexible prior models for the marked Poisson process. In particular, we focus on fully nonparametric model formulations that build the mark density and intensity function from a joint nonparametric mixture, and provide guidelines for straightforward application of these techniques. A key feature of such models is that they can yield flexible inference about the conditional distribution for multivariate marks without requiring specification of a complicated dependence scheme. We address issues relating to choice of the Dirichlet process mixture kernels, and develop methods for prior specification and posterior simulation for full inference about functionals of the marked Poisson process. Moreover, we discuss a method for model checking that can be used to assess and compare goodness of fit of different model specifications under the proposed framework. The methodology is illustrated with simulated and real data sets.

Computing the Bayes Factor from a Markov chain Monte Carlo Simulation of the Posterior Distribution

Abstract: Determining the marginal likelihood from a simulated posterior distribution is central to Bayesian model selection but is computationally challenging. The often-used harmonic mean approximation (HMA) makes no prior assumptions about the character of the distribution but tends to be inconsistent. The Laplace approximation is stable but makes strong, and often inappropriate, assumptions about the shape of the posterior distribution. Here, I argue that the marginal likelihood can be reliably computed from a posterior sample using Lebesgue integration theory in one of two ways: 1) when the HMA integral exists, compute the measure function numerically and analyze the resulting quadrature to control error; 2) compute the measure function numerically for the marginal likelihood integral itself using a space-partitioning tree, followed by quadrature. The first algorithm automatically eliminates the part of the sample that contributes large truncation error in the HMA. Moreover, it provides a simple graphical test for the existence of the HMA integral. The second algorithm uses the posterior sample to assign probability to a partition of the sample space and performs the marginal likelihood integral directly. It uses the posterior sample to discover and tessellate the subset of the sample space that was explored and uses quantiles to compute a representative field value. When integrating directly, this space may be trimmed to remove regions with low probability density and thereby improve accuracy. This second algorithm is consistent for all proper distributions. Error analysis provides some diagnostics on the numerical condition of the results in both cases.

A Bayesian Semiparametric Temporally-Stratified Proportional Hazards Model with Spatial Frailties

Abstract: Incorporating temporal and spatial variation could potentially enhance information gathered from survival data. This paper proposes a Bayesian semiparametric model for capturing spatio-temporal heterogeneity within the proportional hazards framework. The spatial correlation is introduced in the form of county-level frailties. The temporal effect is introduced by considering the stratification of the proportional hazards model, where the time-dependent hazards are indirectly modeled using a probability model for related probability distributions. With this aim, an autoregressive dependent tailfree process is introduced. The full Kullback-Leibler support of the proposed process is provided. The approach is illustrated using simulated and data from the Surveillance Epidemiology and End Results database of the National Cancer Institute on patients in Iowa diagnosed with breast cancer.

The Discrete Innite Logistic Normal Distribution

Abstract: We present the discrete innite logistic normal distribution (DILN), a Bayesian nonparametric prior for mixed membership models. DILN generalizes the hierarchical Dirichlet process (HDP) to model correlation structure between the weights of the atoms at the group level. We derive a representation of DILN as a normalized collection of gamma-distributed random variables and study its statistical properties. We derive a variational inference algorithm for approximate posterior inference. We apply DILN to topic modeling of documents and study its empirical performance on four corpora, comparing performance with the HDP and the correlated topic model (CTM). To compute with large-scale data, we develop a stochastic variational inference algorithm for DILN and compare with similar algorithms for HDP and latent Dirichlet allocation (LDA) on a collection of 350; 000 articles from Nature.

Regularization in Regression: Comparing Bayesian and Frequentist Methods in a Poorly Informative Situation

Abstract: Using a collection of simulated and real benchmarks, we compare Bayesian and frequentist regularization approaches under a low informative constraint when the number of variables is almost equal to the number of observations on simulated and real datasets. This comparison includes new global noninformative approaches for Bayesian variable selection built on Zellner’s g-priors that are similar to Liang et al. (2008). The interest of those calibration-free proposals is discussed. The numerical experiments we present highlight the appeal of Bayesian regularization methods, when compared with non-Bayesian alternatives. They dominate frequentist methods in the sense that they provide smaller prediction errors while selecting the most relevant variables in a parsimonious way.

Posterior Concentration Rates for Infinite Dimensional Exponential Families

Abstract: In this paper we derive adaptive non-parametric rates of concentration of the posterior distributions for the density model on the class of Sobolev and Besov spaces. For this purpose, we build prior models based on wavelet or Fourier expansions of the logarithm of the density. The prior models are not necessarily Gaussian.

Prior effective sample size in conditionally independent hierarchical models

Abstract: Prior effective sample size (ESS) of a Bayesian parametric model was defined by Morita, et al. (2008, Biometrics,64, 595-602). Starting with an e-information prior defined to have the same means and correlations as the prior but to be vague in a suitable sense, the ESS is the required sample size to obtain a hypothetical posterior very close to the prior. In this paper, we present two alternative definitions for the prior ESS that are suitable for a conditionally independent hierarchical model. The two definitions focus on either the first level prior or second level prior. The proposed methods are applied to important examples to verify that each of the two types of prior ESS matches the intuitively obvious answer where it exists. We illustrate the method with applications to several motivating examples, including a single-arm clinical trial to evaluate treatment response probabilities across different disease subtypes, a dose-finding trial based on toxicity in this setting, and a multicenter randomized trial of treatments for affective disorders.

Log-Linear Pool to Combine Prior Distributions: A Suggestion for a Calibration-Based Approach

Abstract: An important issue involved in group decision making is the suitable aggregation of experts’ beliefs about a parameter of interest. Two widely used combination methods are linear and log-linear pools. Yet, a problem arises when the weights have to be selected. This paper provides a general decision-based procedure to obtain the weights in a log-linear pooled prior distribution. The process is based on Kullback-Leibler divergence, which is used as a calibration tool. No information about the parameter of interest is considered before dealing with the experts’ beliefs. Then, a pooled prior distribution is achieved, for which the expected calibration is the best one in the Kullback-Leibler sense. In the absence of other information available to the decision-maker prior to getting experimental data, the methodology generally leads to selection of the most diffuse pooled prior. In most cases, a problem arises from the marginal distribution related to the noninformative prior distribution since it is improper. In these cases, an alternative procedure is proposed. Finally, two applications show how the proposed techniques can be easily applied in practice.

Bayesian Model Selection for Beta Autoregressive Processes

Abstract: We deal with Bayesian inference for Beta autoregressive processes. We restrict our attention to the class of conditionally linear processes. These processes are particularly suitable for forecasting purposes, but are difficult to estimate due to the constraints on the parameter space. We provide a full Bayesian approach to the estimation and include the parameter restrictions in the inference problem by a suitable specification of the prior distributions. Moreover in a Bayesian framework parameter estimation and model choice can be solved simultaneously. In particular we suggest a Markov-Chain Monte Carlo (MCMC) procedure based on a Metropolis-Hastings within Gibbs algorithm and solve the model selection problem following a reversible jump MCMC approach.

Two-pronged Strategy for Using DIC to Compare Selection Models with Non-Ignorable Missing Responses

Abstract: Data with missing responses generated by a non-ignorable missing- ness mechanism can be analysed by jointly modelling the response and a binary variable indicating whether the response is observed or missing. Using a selection model factorisation, the resulting joint model consists of a model of interest and a model of missingness. In the case of non-ignorable missingness, model choice is di-cult because the assumptions about the missingness model are never veriflable from the data at hand. For complete data, the Deviance Information Criterion (DIC) is routinely used for Bayesian model comparison. However, when an anal- ysis includes missing data, DIC can be constructed in difierent ways and its use and interpretation are not straightforward. In this paper, we present a strategy for comparing selection models by combining information from two measures taken from difierent constructions of the DIC. A DIC based on the observed data likeli- hood is used to compare joint models with difierent models of interest but the same model of missingness, and a comparison of models with the same model of interest but difierent models of missingness is carried out using the model of missingness part of a conditional DIC. This strategy is intended for use within a sensitivity analysis that explores the impact of difierent assumptions about the two parts of the model, and is illustrated by examples with simulated missingness and an appli- cation which compares three treatments for depression using data from a clinical trial. We also examine issues relating to the calculation of the DIC based on the observed data likelihood.

Universality of Bayesian Predictions

Abstract: This paper studies the theoretical properties of Bayesian predictions and shows that under minimal conditions we can derive finite sample bounds for the loss incurred using Bayesian predictions under the Kullback-Leibler divergence. In particular, the concept of universality of predictions is discussed and universality is established for Bayesian predictions in a variety of settings. These include predictions under almost arbitrary loss functions, model averaging, predictions in a non-stationary environment and under model misspecification.

Objective Bayesian analysis of a measurement error small area model

Abstract: We consider small area estimation under a nested error linear regression model with measurement errors in the covariates. We propose an objective Bayesian analysis of the model to estimate the finite population means of the small areas. In particular, we derive Jeffreys’ prior for model parameters. We also show that Jeffreys’ prior, which is improper, leads, under very general conditions, to a proper posterior distribution. We have also performed a simulation study where we have compared the Bayes estimates of the finite population means under the Jeffreys’ prior with other Bayesian estimates obtained via the use of the standard flat prior and with non-Bayesian estimates, i.e., the corresponding empirical Bayes estimates and the direct estimates.

Flexible Bayesian Human Fecundity Models

Abstract: Human fecundity is an issue of considerable interest for both epidemiological and clinical audiences, and is dependent upon a couple’s biologic capacity for reproduction coupled with behaviors that place a couple at risk for pregnancy. Bayesian hierarchical models have been proposed to better model the conception probabilities by accounting for the acts of intercourse around the day of ovulation, i.e., during the fertile window. These models can be viewed in the framework of a generalized nonlinear model with an exponential link. However, a fixed choice of link function may not always provide the best fit, leading to potentially biased estimates for probability of conception. Motivated by this, we propose a general class of models for fecundity by relaxing the choice of the link function under the generalized nonlinear model framework. We use a sample from the Oxford Conception Study (OCS) to illustrate the utility and fit of this general class of models for estimating human conception. Our findings reinforce the need for attention to be paid to the choice of link function in modeling conception, as it may bias the estimation of conception probabilities. Various properties of the proposed models are examined and a Markov chain Monte Carlo sampling algorithm was developed for implementing the Bayesian computations. The deviance information criterion measure and logarithm of pseudo marginal likelihood are used for guiding the choice of links. The supplemental material section contains technical details of the proof of the theorem stated in the paper, and contains further simulation results and analysis.

Bayesian Strategies to Assess Uncertainty in Velocity Models

Abstract: Quantifying uncertainty in models derived from observed seismic data is a major issue. In this research we examine the geological structure of the sub-surface using controlled source seismology which gives the data in time and the distance between the acoustic source and the receiver. Inversion tools exist to map these data into a depth model, but a full exploration of the uncertainty of the model is rarely done because robust strategies do not exist for large non-linear complex systems. There are two principal sources of uncertainty: the first comes from the input data which is noisy and band-limited; the second is from the model parameterisation and forward algorithm which approximate the physics to make the problem tractable. To address these issues we propose a Bayesian approach using the Metropolis-Hastings algorithm.

Bayesian Estimation of Log-Normal Means with Finite Quadratic Expected Loss

Abstract: The log-normal distribution is a popular model in biostatistics and other fields of statistics. Bayesian inference on the mean and median of the dis- tribution is problematic because, for many popular choices of the prior for the variance (on the log-scale) parameter, the posterior distribution has no finite mo- ments, leading to Bayes estimators with infinite expected loss for the most common choices of the loss function. We propose a generalized inverse Gaussian prior for the variance parameter, that leads to a log-generalized hyperbolic posterior, for which it is easy to calculate quantiles and moments, provided that they exist. We derive the constraints on the prior parameters that yield finite posterior moments of or- der r. We investigate the choice of prior parameters leading to Bayes estimators with optimal frequentist mean square error. For the estimation of the lognormal mean we show, using simulation, that the Bayes estimator under quadratic loss compares favorably in terms of frequentist mean square error to known estimators.

Using Individual-Level Models for Infectious Disease Spread to Model Spatio-Temporal Combustion Dynamics

Abstract: Individual-level models (ILMs), as defined by Deardon et al. (2010), are a class of models originally designed to model the spread of infectious disease. However, they can also be considered as a tool for modelling the spatio-temporal dynamics of fire. We consider the much simplified problem of modelling the combustion dynamics on a piece of wax paper under relatively controlled conditions. The models are fitted in a Bayesian framework using Markov chain Monte Carlo (MCMC) methods. The focus here is on choosing a model that best fits the combustion pattern.

Bayesian Matching of Unlabeled Point Sets Using Procrustes and Configuration Models

Abstract: The problem of matching unlabeled point sets using Bayesian inference is considered. Two recently proposed models for the likelihood are compared, based on the Procrustes size-and-shape and the full configuration. Bayesian inference is carried out for matching point sets using Markov chain Monte Carlo simulation. An improvement to the existing Procrustes algorithm is proposed which improves convergence rates, using occasional large jumps in the burn-in period. The Procrustes and configuration methods are compared in a simulation study and using real data, where it is of interest to estimate the strengths of matches between protein binding sites. The performance of both methods is generally quite similar, and a connection between the two models is made using a Laplace approximation.

Nonparametric Bayesian Segmentation of a Multivariate Inhomogeneous Space-Time Poisson Process.

Abstract: A nonparametric Bayesian model is proposed for segmenting time-evolving multivariate spatial point process data. An inhomogeneous Poisson process is assumed, with a logistic stick-breaking process (LSBP) used to encourage piecewise-constant spatial Poisson intensities. The LSBP explicitly favors spatially contiguous segments, and infers the number of segments based on the observed data. The temporal dynamics of the segmentation and of the Poisson intensities are modeled with exponential correlation in time, implemented in the form of a first-order autoregressive model for uniformly sampled discrete data, and via a Gaussian process with an exponential kernel for general temporal sampling. We consider and compare two different inference techniques: a Markov chain Monte Carlo sampler, which has relatively high computational complexity; and an approximate and efficient variational Bayesian analysis. The model is demonstrated with a simulated example and a real example of space-time crime events in Cincinnati, Ohio, USA.