Silvia Metelli, Nicholas Heard.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2586--2610.

Abstract:
Monitoring computer network traffic for anomalous behaviour presents an important security challenge. Arrivals of new edges in a network graph represent connections between a client and server pair not previously observed, and in rare cases these might suggest the presence of intruders or malicious implants. We propose a Bayesian model and anomaly detection method for simultaneously characterising existing network structure and modelling likely new edge formation. The method is demonstrated on real computer network authentication data and successfully identifies some machines which are known to be compromised.

Kun Liang.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2452--2482.

Abstract:
Finding differentially expressed genes is a common task in high-throughput transcriptome studies. While traditional statistical methods rank the genes by their test statistics alone, we analyze an RNA sequencing dataset using the auxiliary information of gene length and the test statistics from a related microarray study. Given the auxiliary information, we propose a novel nonparametric empirical Bayes procedure to estimate the posterior probability of differential expression for each gene. We demonstrate the advantage of our procedure in extensive simulation studies and a psoriasis RNA sequencing study. The companion R package calm is available at Bioconductor.

Federico Castelletti, Guido Consonni.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2289--2311.

Abstract:
A signalling pathway is a sequence of chemical reactions initiated by a stimulus which in turn affects a receptor, and then through some intermediate steps cascades down to the final cell response. Based on the technique of flow cytometry, samples of cell-by-cell measurements are collected under each experimental condition, resulting in a collection of interventional data (assuming no latent variables are involved). Usually several external interventions are applied at different points of the pathway, the ultimate aim being the structural recovery of the underlying signalling network which we model as a causal Directed Acyclic Graph (DAG) using intervention calculus. The advantage of using interventional data, rather than purely observational one, is that identifiability of the true data generating DAG is enhanced. More technically a Markov equivalence class of DAGs, whose members are statistically indistinguishable based on observational data alone, can be further decomposed, using additional interventional data, into smaller distinct Interventional Markov equivalence classes. We present a Bayesian methodology for structural learning of Interventional Markov equivalence classes based on observational and interventional samples of multivariate Gaussian observations. Our approach is objective, meaning that it is based on default parameter priors requiring no personal elicitation; some flexibility is however allowed through a tuning parameter which regulates sparsity in the prior on model space. Based on an analytical expression for the marginal likelihood of a given Interventional Essential Graph, and a suitable MCMC scheme, our analysis produces an approximate posterior distribution on the space of Interventional Markov equivalence classes, which can be used to provide uncertainty quantification for features of substantive scientific interest, such as the posterior probability of inclusion of selected edges, or paths.

Carolyn M. Rutter, Jonathan Ozik, Maria DeYoreo, Nicholson Collier.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2189--2212.

Abstract:
Microsimulation models (MSMs) are used to inform policy by predicting population-level outcomes under different scenarios. MSMs simulate individual-level event histories that mark the disease process (such as the development of cancer) and the effect of policy actions (such as screening) on these events. MSMs often have many unknown parameters; calibration is the process of searching the parameter space to select parameters that result in accurate MSM prediction of a wide range of targets. We develop Incremental Mixture Approximate Bayesian Computation (IMABC) for MSM calibration which results in a simulated sample from the posterior distribution of model parameters given calibration targets. IMABC begins with a rejection-based ABC step, drawing a sample of points from the prior distribution of model parameters and accepting points that result in simulated targets that are near observed targets. Next, the sample is iteratively updated by drawing additional points from a mixture of multivariate normal distributions and accepting points that result in accurate predictions. Posterior estimates are obtained by weighting the final set of accepted points to account for the adaptive sampling scheme. We demonstrate IMABC by calibrating CRC-SPIN 2.0, an updated version of a MSM for colorectal cancer (CRC) that has been used to inform national CRC screening guidelines.

Ye Zhang, Zhigang Yao, Patrik Forssén, Torgny Fornstedt.

Source: The Annals of Applied Statistics, Volume 13, Number 4, 2011--2042.

Abstract:
The means to obtain the rate constants of a chemical reaction is a fundamental open problem in both science and the industry. Traditional techniques for finding rate constants require either chemical modifications of the reactants or indirect measurements. The rate constant map method is a modern technique to study binding equilibrium and kinetics in chemical reactions. Finding a rate constant map from biosensor data is an ill-posed inverse problem that is usually solved by regularization. In this work, rather than finding a deterministic regularized rate constant map that does not provide uncertainty quantification of the solution, we develop an adaptive variational Bayesian approach to estimate the distribution of the rate constant map, from which some intrinsic properties of a chemical reaction can be explored, including information about rate constants. Our new approach is more realistic than the existing approaches used for biosensors and allows us to estimate the dynamics of the interactions, which are usually hidden in a deterministic approximate solution. We verify the performance of the new proposed method by numerical simulations, and compare it with the Markov chain Monte Carlo algorithm. The results illustrate that the variational method can reliably capture the posterior distribution in a computationally efficient way. Finally, the developed method is also tested on the real biosensor data (parathyroid hormone), where we provide two novel analysis tools—the thresholding contour map and the high order moment map—to estimate the number of interactions as well as their rate constants.

Bret Zeldow, Vincent Lo Re III, Jason Roy.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1989--2010.

Abstract:
Bayesian Additive Regression Trees (BART) is a flexible machine learning algorithm capable of capturing nonlinearities between an outcome and covariates and interactions among covariates. We extend BART to a semiparametric regression framework in which the conditional expectation of an outcome is a function of treatment, its effect modifiers, and confounders. The confounders are allowed to have unspecified functional form, while treatment and effect modifiers that are directly related to the research question are given a linear form. The result is a Bayesian semiparametric linear regression model where the posterior distribution of the parameters of the linear part can be interpreted as in parametric Bayesian regression. This is useful in situations where a subset of the variables are of substantive interest and the others are nuisance variables that we would like to control for. An example of this occurs in causal modeling with the structural mean model (SMM). Under certain causal assumptions, our method can be used as a Bayesian SMM. Our methods are demonstrated with simulation studies and an application to dataset involving adults with HIV/Hepatitis C coinfection who newly initiate antiretroviral therapy. The methods are available in an R package called semibart.

Youyi Zhang, Jeffrey S. Morris, Shivali Narang Aerry, Arvind U. K. Rao, Veerabhadran Baladandayuthapani.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1957--1988.

Abstract:
Technological innovations have produced large multi-modal datasets that include imaging and multi-platform genomics data. Integrative analyses of such data have the potential to reveal important biological and clinical insights into complex diseases like cancer. In this paper, we present Bayesian approaches for integrative analysis of radiological imaging and multi-platform genomic data, where-in our goals are to simultaneously identify genomic and radiomic, that is, radiology-based imaging markers, along with the latent associations between these two modalities, and to detect the overall prognostic relevance of the combined markers. For this task, we propose Radio-iBAG: Radiomics-based Integrative Bayesian Analysis of Multiplatform Genomic Data , a multi-scale Bayesian hierarchical model that involves several innovative strategies: it incorporates integrative analysis of multi-platform genomic data sets to capture fundamental biological relationships; explores the associations between radiomic markers accompanying genomic information with clinical outcomes; and detects genomic and radiomic markers associated with clinical prognosis. We also introduce the use of sparse Principal Component Analysis (sPCA) to extract a sparse set of approximately orthogonal meta-features each containing information from a set of related individual radiomic features, reducing dimensionality and combining like features. Our methods are motivated by and applied to The Cancer Genome Atlas glioblastoma multiforme data set, where-in we integrate magnetic resonance imaging-based biomarkers along with genomic, epigenomic and transcriptomic data. Our model identifies important magnetic resonance imaging features and the associated genomic platforms that are related with patient survival times.

Chanmin Kim, Michael J. Daniels, Joseph W. Hogan, Christine Choirat, Corwin M. Zigler.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1927--1956.

Abstract:
Emission control technologies installed on power plants are a key feature of many air pollution regulations in the US. While such regulations are predicated on the presumed relationships between emissions, ambient air pollution and human health, many of these relationships have never been empirically verified. The goal of this paper is to develop new statistical methods to quantify these relationships. We frame this problem as one of mediation analysis to evaluate the extent to which the effect of a particular control technology on ambient pollution is mediated through causal effects on power plant emissions. Since power plants emit various compounds that contribute to ambient pollution, we develop new methods for multiple intermediate variables that are measured contemporaneously, may interact with one another, and may exhibit joint mediating effects. Specifically, we propose new methods leveraging two related frameworks for causal inference in the presence of mediating variables: principal stratification and causal mediation analysis. We define principal effects based on multiple mediators, and also introduce a new decomposition of the total effect of an intervention on ambient pollution into the natural direct effect and natural indirect effects for all combinations of mediators. Both approaches are anchored to the same observed-data models, which we specify with Bayesian nonparametric techniques. We provide assumptions for estimating principal causal effects, then augment these with an additional assumption required for causal mediation analysis. The two analyses, interpreted in tandem, provide the first empirical investigation of the presumed causal pathways that motivate important air quality regulatory policies.

Arkaprava Roy, Subhashis Ghosal, Jeffrey Prescott, Kingshuk Roy Choudhury.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1791--1816.

Abstract:
We study possible relations between Alzheimer’s disease progression and the structure of the connectome which is white matter connecting different regions of the brain. Regression models in covariates including age, gender and disease status for the extent of white matter connecting each pair of regions of the brain are proposed. Subject inhomogeneity is also incorporated in the model through random effects with an unknown distribution. As there is a large number of pairs of regions, we also adopt a dimension reduction technique through graphon ( J. Combin. Theory Ser. B 96 (2006) 933–957) functions which reduces the functions of pairs of regions to functions of regions. The connecting graphon functions are considered unknown but the assumed smoothness allows putting priors of low complexity on these functions. We pursue a nonparametric Bayesian approach by assigning a Dirichlet process scale mixture of zero to mean normal prior on the distributions of the random effects and finite random series of tensor products of B-splines priors on the underlying graphon functions. We develop efficient Markov chain Monte Carlo techniques for drawing samples for the posterior distributions using Hamiltonian Monte Carlo (HMC). The proposed Bayesian method overwhelmingly outperforms a competing method based on ANCOVA models in the simulation setup. The proposed Bayesian approach is applied on a dataset of 100 subjects and 83 brain regions and key regions implicated in the changing connectome are identified.

Yiyi Liu, Joshua L. Warren, Hongyu Zhao.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1733--1752.

Abstract:
Understanding the heterogeneity of cells is an important biological question. The development of single-cell RNA-sequencing (scRNA-seq) technology provides high resolution data for such inquiry. A key challenge in scRNA-seq analysis is the high variability of measured RNA expression levels and frequent dropouts (missing values) due to limited input RNA compared to bulk RNA-seq measurement. Existing clustering methods do not perform well for these noisy and zero-inflated scRNA-seq data. In this manuscript we propose a Bayesian hierarchical model, called BasClu, to appropriately characterize important features of scRNA-seq data in order to more accurately cluster cells. We demonstrate the effectiveness of our method with extensive simulation studies and applications to three real scRNA-seq datasets.

Qiwei Li, Xinlei Wang, Faming Liang, Guanghua Xiao.

Source: The Annals of Applied Statistics, Volume 13, Number 3, 1708--1732.

Abstract:
With the advance of imaging technology, digital pathology imaging of tumor tissue slides is becoming a routine clinical procedure for cancer diagnosis. This process produces massive imaging data that capture histological details in high resolution. Recent developments in deep-learning methods have enabled us to identify and classify individual cells from digital pathology images at large scale. Reliable statistical approaches to model the spatial pattern of cells can provide new insight into tumor progression and shed light on the biological mechanisms of cancer. We consider the problem of modeling spatial correlations among three commonly seen cells observed in tumor pathology images. A novel geostatistical marking model with interpretable underlying parameters is proposed in a Bayesian framework. We use auxiliary variable MCMC algorithms to sample from the posterior distribution with an intractable normalizing constant. We demonstrate how this model-based analysis can lead to sharper inferences than ordinary exploratory analyses, by means of application to three benchmark datasets and a case study on the pathology images of $188$ lung cancer patients. The case study shows that the spatial correlation between tumor and stromal cells predicts patient prognosis. This statistical methodology not only presents a new model for characterizing spatial correlations in a multitype spatial point pattern conditioning on the locations of the points, but also provides a new perspective for understanding the role of cell–cell interactions in cancer progression.

Bo Ning, Seonghyun Jeong, Subhashis Ghosal.

Source: Bernoulli, Volume 26, Number 3, 2353--2382.

Abstract:
We study frequentist properties of a Bayesian high-dimensional multivariate linear regression model with correlated responses. The predictors are separated into many groups and the group structure is pre-determined. Two features of the model are unique: (i) group sparsity is imposed on the predictors; (ii) the covariance matrix is unknown and its dimensions can also be high. We choose a product of independent spike-and-slab priors on the regression coefficients and a new prior on the covariance matrix based on its eigendecomposition. Each spike-and-slab prior is a mixture of a point mass at zero and a multivariate density involving the $ell_{2,1}$-norm. We first obtain the posterior contraction rate, the bounds on the effective dimension of the model with high posterior probabilities. We then show that the multivariate regression coefficients can be recovered under certain compatibility conditions. Finally, we quantify the uncertainty for the regression coefficients with frequentist validity through a Bernstein–von Mises type theorem. The result leads to selection consistency for the Bayesian method. We derive the posterior contraction rate using the general theory by constructing a suitable test from the first principle using moment bounds for certain likelihood ratios. This leads to posterior concentration around the truth with respect to the average Rényi divergence of order $1/2$. This technique of obtaining the required tests for posterior contraction rate could be useful in many other problems.

Ester Mariucci, Kolyan Ray, Botond Szabó.

Source: Bernoulli, Volume 26, Number 2, 1070--1097.

Abstract:
The estimation of a log-concave density on $mathbb{R}$ is a canonical problem in the area of shape-constrained nonparametric inference. We present a Bayesian nonparametric approach to this problem based on an exponentiated Dirichlet process mixture prior and show that the posterior distribution converges to the log-concave truth at the (near-) minimax rate in Hellinger distance. Our proof proceeds by establishing a general contraction result based on the log-concave maximum likelihood estimator that prevents the need for further metric entropy calculations. We further present computationally more feasible approximations and both an empirical and hierarchical Bayes approach. All priors are illustrated numerically via simulations.

Keisuke Yano, Kengo Kato.

Source: Bernoulli, Volume 26, Number 1, 616--641.

Abstract:
In this paper, we study frequentist coverage errors of Bayesian credible sets for an approximately linear regression model with (moderately) high dimensional regressors, where the dimension of the regressors may increase with but is smaller than the sample size. Specifically, we consider quasi-Bayesian inference on the slope vector under the quasi-likelihood with Gaussian error distribution. Under this setup, we derive finite sample bounds on frequentist coverage errors of Bayesian credible rectangles. Derivation of those bounds builds on a novel Berry–Esseen type bound on quasi-posterior distributions and recent results on high-dimensional CLT on hyperrectangles. We use this general result to quantify coverage errors of Castillo–Nickl and $L^{infty}$-credible bands for Gaussian white noise models, linear inverse problems, and (possibly non-Gaussian) nonparametric regression models. In particular, we show that Bayesian credible bands for those nonparametric models have coverage errors decaying polynomially fast in the sample size, implying advantages of Bayesian credible bands over confidence bands based on extreme value theory.

Zhi-Qiang Wang, Nian-Sheng Tang.

Source: Bayesian Analysis, Volume 15, Number 2, 579--604.

Abstract:
Bayesian inference on quantile regression (QR) model with mixed discrete and non-ignorable missing covariates is conducted by reformulating QR model as a hierarchical structure model. A probit regression model is adopted to specify missing covariate mechanism. A hybrid algorithm combining the Gibbs sampler and the Metropolis-Hastings algorithm is developed to simultaneously produce Bayesian estimates of unknown parameters and latent variables as well as their corresponding standard errors. Bayesian variable selection method is proposed to recognize significant covariates. A Bayesian local influence procedure is presented to assess the effect of minor perturbations to the data, priors and sampling distributions on posterior quantities of interest. Several simulation studies and an example are presented to illustrate the proposed methodologies.

Kurtis Shuler, Marilou Sison-Mangus, Juhee Lee.

Source: Bayesian Analysis, Volume 15, Number 2, 559--578.

Abstract:
We propose a Bayesian sparse multivariate regression method to model the relationship between microbe abundance and environmental factors for microbiome data. We model abundance counts of operational taxonomic units (OTUs) with a negative binomial distribution and relate covariates to the counts through regression. Extending conventional nonlocal priors, we construct asymmetric nonlocal priors for regression coefficients to efficiently identify relevant covariates and their effect directions. We build a hierarchical model to facilitate pooling of information across OTUs that produces parsimonious results with improved accuracy. We present simulation studies that compare variable selection performance under the proposed model to those under Bayesian sparse regression models with asymmetric and symmetric local priors and two frequentist models. The simulations show the proposed model identifies important covariates and yields coefficient estimates with favorable accuracy compared with the alternatives. The proposed model is applied to analyze an ocean microbiome dataset collected over time to study the association of harmful algal bloom conditions with microbial communities.

Jami J. Mulgrave, Subhashis Ghosal.

Source: Bayesian Analysis, Volume 15, Number 2, 449--475.

Abstract:
Gaussian graphical models have been used to study intrinsic dependence among several variables, but the Gaussianity assumption may be restrictive in many applications. A nonparanormal graphical model is a semiparametric generalization for continuous variables where it is assumed that the variables follow a Gaussian graphical model only after some unknown smooth monotone transformations on each of them. We consider a Bayesian approach in the nonparanormal graphical model by putting priors on the unknown transformations through a random series based on B-splines where the coefficients are ordered to induce monotonicity. A truncated normal prior leads to partial conjugacy in the model and is useful for posterior simulation using Gibbs sampling. On the underlying precision matrix of the transformed variables, we consider a spike-and-slab prior and use an efficient posterior Gibbs sampling scheme. We use the Bayesian Information Criterion to choose the hyperparameters for the spike-and-slab prior. We present a posterior consistency result on the underlying transformation and the precision matrix. We study the numerical performance of the proposed method through an extensive simulation study and finally apply the proposed method on a real data set.

Philippe Gagnon, Alain Desgagné, Mylène Bédard.

Source: Bayesian Analysis, Volume 15, Number 2, 389--414.

Abstract:
Linear regression is ubiquitous in statistical analysis. It is well understood that conflicting sources of information may contaminate the inference when the classical normality of errors is assumed. The contamination caused by the light normal tails follows from an undesirable effect: the posterior concentrates in an area in between the different sources with a large enough scaling to incorporate them all. The theory of conflict resolution in Bayesian statistics (O’Hagan and Pericchi (2012)) recommends to address this problem by limiting the impact of outliers to obtain conclusions consistent with the bulk of the data. In this paper, we propose a model with super heavy-tailed errors to achieve this. We prove that it is wholly robust, meaning that the impact of outliers gradually vanishes as they move further and further away from the general trend. The super heavy-tailed density is similar to the normal outside of the tails, which gives rise to an efficient estimation procedure. In addition, estimates are easily computed. This is highlighted via a detailed user guide, where all steps are explained through a simulated case study. The performance is shown using simulation. All required code is given.

Andrés F. Barrientos, Víctor Peña.

Source: Bayesian Analysis, Volume 15, Number 2, 363--388.

Abstract:
In this article, we present data-subsetting algorithms that allow for the approximate and scalable implementation of the Bayesian bootstrap. They are analogous to two existing algorithms in the frequentist literature: the bag of little bootstraps (Kleiner et al., 2014) and the subsampled double bootstrap (Sengupta et al., 2016). Our algorithms have appealing theoretical and computational properties that are comparable to those of their frequentist counterparts. Additionally, we provide a strategy for performing lossless inference for a class of functionals of the Bayesian bootstrap and briefly introduce extensions to the Dirichlet Process.

Kelly C. M. Gonçalves, Hélio S. Migon, Leonardo S. Bastos.

Source: Bayesian Analysis, Volume 15, Number 2, 335--362.

Abstract:
The paper introduces a new class of models, named dynamic quantile linear models, which combines dynamic linear models with distribution-free quantile regression producing a robust statistical method. Bayesian estimation for the dynamic quantile linear model is performed using an efficient Markov chain Monte Carlo algorithm. The paper also proposes a fast sequential procedure suited for high-dimensional predictive modeling with massive data, where the generating process is changing over time. The proposed model is evaluated using synthetic and well-known time series data. The model is also applied to predict annual incidence of tuberculosis in the state of Rio de Janeiro and compared with global targets set by the World Health Organization.

Aliaksandr Hubin, Geir Storvik, Florian Frommlet.

Source: Bayesian Analysis, Volume 15, Number 1, 263--333.

Abstract:
Logic regression was developed more than a decade ago as a tool to construct predictors from Boolean combinations of binary covariates. It has been mainly used to model epistatic effects in genetic association studies, which is very appealing due to the intuitive interpretation of logic expressions to describe the interaction between genetic variations. Nevertheless logic regression has (partly due to computational challenges) remained less well known than other approaches to epistatic association mapping. Here we will adapt an advanced evolutionary algorithm called GMJMCMC (Genetically modified Mode Jumping Markov Chain Monte Carlo) to perform Bayesian model selection in the space of logic regression models. After describing the algorithmic details of GMJMCMC we perform a comprehensive simulation study that illustrates its performance given logic regression terms of various complexity. Specifically GMJMCMC is shown to be able to identify three-way and even four-way interactions with relatively large power, a level of complexity which has not been achieved by previous implementations of logic regression. We apply GMJMCMC to reanalyze QTL (quantitative trait locus) mapping data for Recombinant Inbred Lines in Arabidopsis thaliana and from a backcross population in Drosophila where we identify several interesting epistatic effects. The method is implemented in an R package which is available on github.

Fangzheng Xie, Yanxun Xu.

Source: Bayesian Analysis, Volume 15, Number 1, 159--186.

Abstract:
We propose a kernel mixture of polynomials prior for Bayesian nonparametric regression. The regression function is modeled by local averages of polynomials with kernel mixture weights. We obtain the minimax-optimal contraction rate of the full posterior distribution up to a logarithmic factor by estimating metric entropies of certain function classes. Under the assumption that the degree of the polynomials is larger than the unknown smoothness level of the true function, the posterior contraction behavior can adapt to this smoothness level provided an upper bound is known. We also provide a frequentist sieve maximum likelihood estimator with a near-optimal convergence rate. We further investigate the application of the kernel mixture of polynomials to partial linear models and obtain both the near-optimal rate of contraction for the nonparametric component and the Bernstein-von Mises limit (i.e., asymptotic normality) of the parametric component. The proposed method is illustrated with numerical examples and shows superior performance in terms of computational efficiency, accuracy, and uncertainty quantification compared to the local polynomial regression, DiceKriging, and the robust Gaussian stochastic process.

Antony Overstall, James McGree.

Source: Bayesian Analysis, Volume 15, Number 1, 103--131.

Abstract:
A Bayesian design is given by maximising an expected utility over a design space. The utility is chosen to represent the aim of the experiment and its expectation is taken with respect to all unknowns: responses, parameters and/or models. Although straightforward in principle, there are several challenges to finding Bayesian designs in practice. Firstly, the utility and expected utility are rarely available in closed form and require approximation. Secondly, the design space can be of high-dimensionality. In the case of intractable likelihood models, these problems are compounded by the fact that the likelihood function, whose evaluation is required to approximate the expected utility, is not available in closed form. A strategy is proposed to find Bayesian designs for intractable likelihood models. It relies on the development of an automatic, auxiliary modelling approach, using multivariate Gaussian process emulators, to approximate the likelihood function. This is then combined with a copula-based approach to approximate the marginal likelihood (a quantity commonly required to evaluate many utility functions). These approximations are demonstrated on examples of stochastic process models involving experimental aims of both parameter estimation and model comparison.

Qingpo Cai, Jian Kang, Tianwei Yu.

Source: Bayesian Analysis, Volume 15, Number 1, 79--102.

Abstract:
Selecting informative nodes over large-scale networks becomes increasingly important in many research areas. Most existing methods focus on the local network structure and incur heavy computational costs for the large-scale problem. In this work, we propose a novel prior model for Bayesian network marker selection in the generalized linear model (GLM) framework: the Thresholded Graph Laplacian Gaussian (TGLG) prior, which adopts the graph Laplacian matrix to characterize the conditional dependence between neighboring markers accounting for the global network structure. Under mild conditions, we show the proposed model enjoys the posterior consistency with a diverging number of edges and nodes in the network. We also develop a Metropolis-adjusted Langevin algorithm (MALA) for efficient posterior computation, which is scalable to large-scale networks. We illustrate the superiorities of the proposed method compared with existing alternatives via extensive simulation studies and an analysis of the breast cancer gene expression dataset in the Cancer Genome Atlas (TCGA).

Matthew R. Williams, Terrance D. Savitsky.

Source: Bayesian Analysis, Volume 15, Number 1, 57--77.

Abstract:
An informative sampling design leads to unit inclusion probabilities that are correlated with the response variable of interest. However, multistage sampling designs may also induce higher order dependencies, which are ignored in the literature when establishing consistency of estimators for survey data under a condition requiring asymptotic independence among the unit inclusion probabilities. This paper constructs new theoretical conditions that guarantee that the pseudo-posterior, which uses sampling weights based on first order inclusion probabilities to exponentiate the likelihood, is consistent not only for survey designs which have asymptotic factorization, but also for survey designs that induce residual or unattenuated dependence among sampled units. The use of the survey-weighted pseudo-posterior, together with our relaxed requirements for the survey design, establish a wide variety of analysis models that can be applied to a broad class of survey data sets. Using the complex sampling design of the National Survey on Drug Use and Health, we demonstrate our new theoretical result on multistage designs characterized by a cluster sampling step that expresses within-cluster dependence. We explore the impact of multistage designs and order based sampling.

Nicolas Garcia Trillos, Daniel Sanz-Alonso.

Source: Bayesian Analysis, Volume 15, Number 1, 29--56.

Abstract:
The Bayesian update can be viewed as a variational problem by characterizing the posterior as the minimizer of a functional. The variational viewpoint is far from new and is at the heart of popular methods for posterior approximation. However, some of its consequences seem largely unexplored. We focus on the following one: defining the posterior as the minimizer of a functional gives a natural path towards the posterior by moving in the direction of steepest descent of the functional. This idea is made precise through the theory of gradient flows, allowing to bring new tools to the study of Bayesian models and algorithms. Since the posterior may be characterized as the minimizer of different functionals, several variational formulations may be considered. We study three of them and their three associated gradient flows. We show that, in all cases, the rate of convergence of the flows to the posterior can be bounded by the geodesic convexity of the functional to be minimized. Each gradient flow naturally suggests a nonlinear diffusion with the posterior as invariant distribution. These diffusions may be discretized to build proposals for Markov chain Monte Carlo (MCMC) algorithms. By construction, the diffusions are guaranteed to satisfy a certain optimality condition, and rates of convergence are given by the convexity of the functionals. We use this observation to propose a criterion for the choice of metric in Riemannian MCMC methods.

Matthew Moores, Geoff Nicholls, Anthony Pettitt, Kerrie Mengersen.

Source: Bayesian Analysis, Volume 15, Number 1, 1--27.

Abstract:
The inverse temperature parameter of the Potts model governs the strength of spatial cohesion and therefore has a major influence over the resulting model fit. A difficulty arises from the dependence of an intractable normalising constant on the value of this parameter and thus there is no closed-form solution for sampling from the posterior distribution directly. There is a variety of computational approaches for sampling from the posterior without evaluating the normalising constant, including the exchange algorithm and approximate Bayesian computation (ABC). A serious drawback of these algorithms is that they do not scale well for models with a large state space, such as images with a million or more pixels. We introduce a parametric surrogate model, which approximates the score function using an integral curve. Our surrogate model incorporates known properties of the likelihood, such as heteroskedasticity and critical temperature. We demonstrate this method using synthetic data as well as remotely-sensed imagery from the Landsat-8 satellite. We achieve up to a hundredfold improvement in the elapsed runtime, compared to the exchange algorithm or ABC. An open-source implementation of our algorithm is available in the R package bayesImageS .

Jeong Eun Lee, Geoff K. Nicholls, Robin J. Ryder.

Source: Bayesian Analysis, Volume 14, Number 4, 1245--1269.

Abstract:
We develop and apply two calibration procedures for checking the coverage of approximate Bayesian credible sets, including intervals estimated using Monte Carlo methods. The user has an ideal prior and likelihood, but generates a credible set for an approximate posterior based on some approximate prior and likelihood. We estimate the realised posterior coverage achieved by the approximate credible set. This is the coverage of the unknown “true” parameter if the data are a realisation of the user’s ideal observation model conditioned on the parameter, and the parameter is a draw from the user’s ideal prior. In one approach we estimate the posterior coverage at the data by making a semi-parametric logistic regression of binary coverage outcomes on simulated data against summary statistics evaluated on simulated data. In another we use Importance Sampling from the approximate posterior, windowing simulated data to fall close to the observed data. We illustrate our methods on four examples.

Nadja Klein, Michael Stanley Smith.

Source: Bayesian Analysis, Volume 14, Number 4, 1143--1171.

Abstract:
We show how to extract the implicit copula of a response vector from a Bayesian regularized regression smoother with Gaussian disturbances. The copula can be used to compare smoothers that employ different shrinkage priors and function bases. We illustrate with three popular choices of shrinkage priors—a pairwise prior, the horseshoe prior and a g prior augmented with a point mass as employed for Bayesian variable selection—and both univariate and multivariate function bases. The implicit copulas are high-dimensional, have flexible dependence structures that are far from that of a Gaussian copula, and are unavailable in closed form. However, we show how they can be evaluated by first constructing a Gaussian copula conditional on the regularization parameters, and then integrating over these. Combined with non-parametric margins the regularized smoothers can be used to model the distribution of non-Gaussian univariate responses conditional on the covariates. Efficient Markov chain Monte Carlo schemes for evaluating the copula are given for this case. Using both simulated and real data, we show how such copula smoothing models can improve the quality of resulting function estimates and predictive distributions.

Guillaume Kon Kam King, Antonio Canale, Matteo Ruggiero.

Source: Bayesian Analysis, Volume 14, Number 4, 1121--1141.

Abstract:
Motivated by the problem of forecasting demand and offer curves, we introduce a class of nonparametric dynamic models with locally-autoregressive behaviour, and provide a full inferential strategy for forecasting time series of piecewise-constant non-decreasing functions over arbitrary time horizons. The model is induced by a non Markovian system of interacting particles whose evolution is governed by a resampling step and a drift mechanism. The former is based on a global interaction and accounts for the volatility of the functional time series, while the latter is determined by a neighbourhood-based interaction with the past curves and accounts for local trend behaviours, separating these from pure noise. We discuss the implementation of the model for functional forecasting by combining a population Monte Carlo and a semi-automatic learning approach to approximate Bayesian computation which require limited tuning. We validate the inference method with a simulation study, and carry out predictive inference on a real dataset on the Italian natural gas market.

Gemma E. Moran, Veronika Ročková, Edward I. George.

Source: Bayesian Analysis, Volume 14, Number 4, 1091--1119.

Abstract:
Consider the problem of high dimensional variable selection for the Gaussian linear model when the unknown error variance is also of interest. In this paper, we show that the use of conjugate shrinkage priors for Bayesian variable selection can have detrimental consequences for such variance estimation. Such priors are often motivated by the invariance argument of Jeffreys (1961). Revisiting this work, however, we highlight a caveat that Jeffreys himself noticed; namely that biased estimators can result from inducing dependence between parameters a priori . In a similar way, we show that conjugate priors for linear regression, which induce prior dependence, can lead to such underestimation in the Bayesian high-dimensional regression setting. Following Jeffreys, we recommend as a remedy to treat regression coefficients and the error variance as independent a priori . Using such an independence prior framework, we extend the Spike-and-Slab Lasso of Ročková and George (2018) to the unknown variance case. This extended procedure outperforms both the fixed variance approach and alternative penalized likelihood methods on simulated data. On the protein activity dataset of Clyde and Parmigiani (1998), the Spike-and-Slab Lasso with unknown variance achieves lower cross-validation error than alternative penalized likelihood methods, demonstrating the gains in predictive accuracy afforded by simultaneous error variance estimation. The unknown variance implementation of the Spike-and-Slab Lasso is provided in the publicly available R package SSLASSO (Ročková and Moran, 2017).

Claudia Kirch, Matthew C. Edwards, Alexander Meier, Renate Meyer.

Source: Bayesian Analysis, Volume 14, Number 4, 1037--1073.

Abstract:
Nonparametric Bayesian inference has seen a rapid growth over the last decade but only few nonparametric Bayesian approaches to time series analysis have been developed. Most existing approaches use Whittle’s likelihood for Bayesian modelling of the spectral density as the main nonparametric characteristic of stationary time series. It is known that the loss of efficiency using Whittle’s likelihood can be substantial. On the other hand, parametric methods are more powerful than nonparametric methods if the observed time series is close to the considered model class but fail if the model is misspecified. Therefore, we suggest a nonparametric correction of a parametric likelihood that takes advantage of the efficiency of parametric models while mitigating sensitivities through a nonparametric amendment. We use a nonparametric Bernstein polynomial prior on the spectral density with weights induced by a Dirichlet process and prove posterior consistency for Gaussian stationary time series. Bayesian posterior computations are implemented via an MH-within-Gibbs sampler and the performance of the nonparametrically corrected likelihood for Gaussian time series is illustrated in a simulation study and in three astronomy applications, including estimating the spectral density of gravitational wave data from the Advanced Laser Interferometer Gravitational-wave Observatory (LIGO).

Miguel de Carvalho, Garritt L. Page, Bradley J. Barney.

Source: Bayesian Analysis, Volume 14, Number 4, 1013--1036.

Abstract:
We provide a geometric interpretation to Bayesian inference that allows us to introduce a natural measure of the level of agreement between priors, likelihoods, and posteriors. The starting point for the construction of our geometry is the observation that the marginal likelihood can be regarded as an inner product between the prior and the likelihood. A key concept in our geometry is that of compatibility, a measure which is based on the same construction principles as Pearson correlation, but which can be used to assess how much the prior agrees with the likelihood, to gauge the sensitivity of the posterior to the prior, and to quantify the coherency of the opinions of two experts. Estimators for all the quantities involved in our geometric setup are discussed, which can be directly computed from the posterior simulation output. Some examples are used to illustrate our methods, including data related to on-the-job drug usage, midge wing length, and prostate cancer.

Jon Cockayne, Chris J. Oates, Ilse C.F. Ipsen, Mark Girolami.

Source: Bayesian Analysis, Volume 14, Number 3, 937--1012.

Abstract:
A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about, for example, the magnitude of the error. In this paper we propose a novel statistical model for this error, set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging.

Muteb Alharthi, Theodore Kypraios, Philip D. O’Neill.

Source: Bayesian Analysis, Volume 14, Number 3, 927--956.

Abstract:
We consider the problem of model choice for stochastic epidemic models given partial observation of a disease outbreak through time. Our main focus is on the use of Bayes factors. Although Bayes factors have appeared in the epidemic modelling literature before, they can be hard to compute and little attention has been given to fundamental questions concerning their utility. In this paper we derive analytic expressions for Bayes factors given complete observation through time, which suggest practical guidelines for model choice problems. We adapt the power posterior method for computing Bayes factors so as to account for missing data and apply this approach to partially observed epidemics. For comparison, we also explore the use of a deviance information criterion for missing data scenarios. The methods are illustrated via examples involving both simulated and real data.

Brian Neelon.

Source: Bayesian Analysis, Volume 14, Number 3, 849--875.

Abstract:
Motivated by a study examining spatiotemporal patterns in inpatient hospitalizations, we propose an efficient Bayesian approach for fitting zero-inflated negative binomial models. To facilitate posterior sampling, we introduce a set of latent variables that are represented as scale mixtures of normals, where the precision terms follow independent Pólya-Gamma distributions. Conditional on the latent variables, inference proceeds via straightforward Gibbs sampling. For fixed-effects models, our approach is comparable to existing methods. However, our model can accommodate more complex data structures, including multivariate and spatiotemporal data, settings in which current approaches often fail due to computational challenges. Using simulation studies, we highlight key features of the method and compare its performance to other estimation procedures. We apply the approach to a spatiotemporal analysis examining the number of annual inpatient admissions among United States veterans with type 2 diabetes.

Ioannis Ntzoufras, Claudia Tarantola, Monia Lupparelli.

Source: Bayesian Analysis, Volume 14, Number 3, 797--823.

Abstract:
We introduce a novel Bayesian approach for quantitative learning for graphical log-linear marginal models. These models belong to curved exponential families that are difficult to handle from a Bayesian perspective. The likelihood cannot be analytically expressed as a function of the marginal log-linear interactions, but only in terms of cell counts or probabilities. Posterior distributions cannot be directly obtained, and Markov Chain Monte Carlo (MCMC) methods are needed. Finally, a well-defined model requires parameter values that lead to compatible marginal probabilities. Hence, any MCMC should account for this important restriction. We construct a fully automatic and efficient MCMC strategy for quantitative learning for such models that handles these problems. While the prior is expressed in terms of the marginal log-linear interactions, we build an MCMC algorithm that employs a proposal on the probability parameter space. The corresponding proposal on the marginal log-linear interactions is obtained via parameter transformation. We exploit a conditional conjugate setup to build an efficient proposal on probability parameters. The proposed methodology is illustrated by a simulation study and a real dataset.

Luis Gutiérrez, Andrés F. Barrientos, Jorge González, Daniel Taylor-Rodríguez.

Source: Bayesian Analysis, Volume 14, Number 2, 649--675.

Abstract:
We propose a Bayesian nonparametric strategy to test for differences between a control group and several treatment regimes. Most of the existing tests for this type of comparison are based on the differences between location parameters. In contrast, our approach identifies differences across the entire distribution, avoids strong modeling assumptions over the distributions for each treatment, and accounts for multiple testing through the prior distribution on the space of hypotheses. The proposal is compared to other commonly used hypothesis testing procedures under simulated scenarios. Two real applications are also analyzed with the proposed methodology.

Marko Järvenpää, Michael U. Gutmann, Arijus Pleska, Aki Vehtari, Pekka Marttinen.

Source: Bayesian Analysis, Volume 14, Number 2, 595--622.

Abstract:
Approximate Bayesian computation (ABC) is a method for Bayesian inference when the likelihood is unavailable but simulating from the model is possible. However, many ABC algorithms require a large number of simulations, which can be costly. To reduce the computational cost, Bayesian optimisation (BO) and surrogate models such as Gaussian processes have been proposed. Bayesian optimisation enables one to intelligently decide where to evaluate the model next but common BO strategies are not designed for the goal of estimating the posterior distribution. Our paper addresses this gap in the literature. We propose to compute the uncertainty in the ABC posterior density, which is due to a lack of simulations to estimate this quantity accurately, and define a loss function that measures this uncertainty. We then propose to select the next evaluation location to minimise the expected loss. Experiments show that the proposed method often produces the most accurate approximations as compared to common BO strategies.

Quan Zhou, Yongtao Guan.

Source: Bayesian Analysis, Volume 14, Number 2, 573--594.

Abstract:
Bayesian variable selection regression (BVSR) is able to jointly analyze genome-wide genetic datasets, but the slow computation via Markov chain Monte Carlo (MCMC) hampered its wide-spread usage. Here we present a novel iterative method to solve a special class of linear systems, which can increase the speed of the BVSR model-fitting tenfold. The iterative method hinges on the complex factorization of the sum of two matrices and the solution path resides in the complex domain (instead of the real domain). Compared to the Gauss-Seidel method, the complex factorization converges almost instantaneously and its error is several magnitude smaller than that of the Gauss-Seidel method. More importantly, the error is always within the pre-specified precision while the Gauss-Seidel method is not. For large problems with thousands of covariates, the complex factorization is 10–100 times faster than either the Gauss-Seidel method or the direct method via the Cholesky decomposition. In BVSR, one needs to repetitively solve large penalized regression systems whose design matrices only change slightly between adjacent MCMC steps. This slight change in design matrix enables the adaptation of the iterative complex factorization method. The computational innovation will facilitate the wide-spread use of BVSR in reanalyzing genome-wide association datasets.

Alberto Cassese, Weixuan Zhu, Michele Guindani, Marina Vannucci.

Source: Bayesian Analysis, Volume 14, Number 2, 553--572.

Abstract:
In many applications, investigators monitor processes that vary in space and time, with the goal of identifying temporally persistent and spatially localized departures from a baseline or “normal” behavior. In this manuscript, we consider the monitoring of pneumonia and influenza (P&I) mortality, to detect influenza outbreaks in the continental United States, and propose a Bayesian nonparametric model selection approach to take into account the spatio-temporal dependence of outbreaks. More specifically, we introduce a zero-inflated conditionally identically distributed species sampling prior which allows borrowing information across time and to assign data to clusters associated to either a null or an alternate process. Spatial dependences are accounted for by means of a Markov random field prior, which allows to inform the selection based on inferences conducted at nearby locations. We show how the proposed modeling framework performs in an application to the P&I mortality data and in a simulation study, and compare with common threshold methods for detecting outbreaks over time, with more recent Markov switching based models, and with spike-and-slab Bayesian nonparametric priors that do not take into account spatio-temporal dependence.

Joris Mulder, Jean-Paul Fox.

Source: Bayesian Analysis, Volume 14, Number 2, 521--552.

Abstract:
The intraclass correlation plays a central role in modeling hierarchically structured data, such as educational data, panel data, or group-randomized trial data. It represents relevant information concerning the between-group and within-group variation. Methods for Bayesian hypothesis tests concerning the intraclass correlation are proposed to improve decision making in hierarchical data analysis and to assess the grouping effect across different group categories. Estimation and testing methods for the intraclass correlation coefficient are proposed under a marginal modeling framework where the random effects are integrated out. A class of stretched beta priors is proposed on the intraclass correlations, which is equivalent to shifted $F$ priors for the between groups variances. Through a parameter expansion it is shown that this prior is conditionally conjugate under the marginal model yielding efficient posterior computation. A special improper case results in accurate coverage rates of the credible intervals even for minimal sample size and when the true intraclass correlation equals zero. Bayes factor tests are proposed for testing multiple precise and order hypotheses on intraclass correlations. These tests can be used when prior information about the intraclass correlations is available or absent. For the noninformative case, a generalized fractional Bayes approach is developed. The method enables testing the presence and strength of grouped data structures without introducing random effects. The methodology is applied to a large-scale survey study on international mathematics achievement at fourth grade to test the heterogeneity in the clustering of students in schools across countries and assessment cycles.

Benjamin Letham, Brian Karrer, Guilherme Ottoni, Eytan Bakshy.

Source: Bayesian Analysis, Volume 14, Number 2, 495--519.

Abstract:
Randomized experiments are the gold standard for evaluating the effects of changes to real-world systems. Data in these tests may be difficult to collect and outcomes may have high variance, resulting in potentially large measurement error. Bayesian optimization is a promising technique for efficiently optimizing multiple continuous parameters, but existing approaches degrade in performance when the noise level is high, limiting its applicability to many randomized experiments. We derive an expression for expected improvement under greedy batch optimization with noisy observations and noisy constraints, and develop a quasi-Monte Carlo approximation that allows it to be efficiently optimized. Simulations with synthetic functions show that optimization performance on noisy, constrained problems outperforms existing methods. We further demonstrate the effectiveness of the method with two real-world experiments conducted at Facebook: optimizing a ranking system, and optimizing server compiler flags.

Suprateek Kundu, Bani K. Mallick, Veera Baladandayuthapani.

Source: Bayesian Analysis, Volume 14, Number 2, 449--476.

Abstract:
There has been an intense development in the Bayesian graphical model literature over the past decade; however, most of the existing methods are restricted to moderate dimensions. We propose a novel graphical model selection approach for large dimensional settings where the dimension increases with the sample size, by decoupling model fitting and covariance selection. First, a full model based on a complete graph is fit under a novel class of mixtures of inverse–Wishart priors, which induce shrinkage on the precision matrix under an equivalence with Cholesky-based regularization, while enabling conjugate updates. Subsequently, a post-fitting model selection step uses penalized joint credible regions to perform model selection. This allows our methods to be computationally feasible for large dimensional settings using a combination of straightforward Gibbs samplers and efficient post-fitting inferences. Theoretical guarantees in terms of selection consistency are also established. Simulations show that the proposed approach compares favorably with competing methods, both in terms of accuracy metrics and computation times. We apply this approach to a cancer genomics data example.

Luis Gutiérrez, Eduardo Gutiérrez-Peña, Ramsés H. Mena.

Source: Bayesian Analysis, Volume 14, Number 2, 427--447.

Abstract:
This work presents a Bayesian predictive approach to statistical shape analysis. A modeling strategy that starts with a Gaussian distribution on the configuration space, and then removes the effects of location, rotation and scale, is studied. This boils down to an application of the projected normal distribution to model the configurations in the shape space, which together with certain identifiability constraints, facilitates parameter interpretation. Having better control over the parameters allows us to generalize the model to a regression setting where the effect of predictors on shapes can be considered. The methodology is illustrated and tested using both simulated scenarios and a real data set concerning eight anatomical landmarks on a sagittal plane of the corpus callosum in patients with autism and in a group of controls.

Haolun Shi, Guosheng Yin.

Source: Bayesian Analysis, Volume 14, Number 2, 399--425.

Abstract:
Bayesian approaches to phase II clinical trial designs are usually based on the posterior distribution of the parameter of interest and calibration of certain threshold for decision making. If the posterior probability is computed and assessed in a sequential manner, the design may involve the problem of multiplicity, which, however, is often a neglected aspect in Bayesian trial designs. To effectively maintain the overall type I error rate, we propose solutions to the problem of multiplicity for Bayesian sequential designs and, in particular, the determination of the cutoff boundaries for the posterior probabilities. We present both theoretical and numerical methods for finding the optimal posterior probability boundaries with $alpha$ -spending functions that mimic those of the frequentist group sequential designs. The theoretical approach is based on the asymptotic properties of the posterior probability, which establishes a connection between the Bayesian trial design and the frequentist group sequential method. The numerical approach uses a sandwich-type searching algorithm, which immensely reduces the computational burden. We apply least-square fitting to find the $alpha$ -spending function closest to the target. We discuss the application of our method to single-arm and double-arm cases with binary and normal endpoints, respectively, and provide a real trial example for each case.

Daniela Pauger, Helga Wagner.

Source: Bayesian Analysis, Volume 14, Number 2, 341--369.

Abstract:
We propose a Bayesian approach to obtain a sparse representation of the effect of a categorical predictor in regression type models. As this effect is captured by a group of level effects, sparsity cannot only be achieved by excluding single irrelevant level effects or the whole group of effects associated to this predictor but also by fusing levels which have essentially the same effect on the response. To achieve this goal, we propose a prior which allows for almost perfect as well as almost zero dependence between level effects a priori. This prior can alternatively be obtained by specifying spike and slab prior distributions on all effect differences associated to this categorical predictor. We show how restricted fusion can be implemented and develop an efficient MCMC (Markov chain Monte Carlo) method for posterior computation. The performance of the proposed method is investigated on simulated data and we illustrate its application on real data from EU-SILC (European Union Statistics on Income and Living Conditions).

Daniele Durante, Tommaso Rigon.

Source: Statistical Science, Volume 34, Number 3, 472--485.

Abstract:
Variational Bayes (VB) is a common strategy for approximate Bayesian inference, but simple methods are only available for specific classes of models including, in particular, representations having conditionally conjugate constructions within an exponential family. Models with logit components are an apparently notable exception to this class, due to the absence of conjugacy among the logistic likelihood and the Gaussian priors for the coefficients in the linear predictor. To facilitate approximate inference within this widely used class of models, Jaakkola and Jordan ( Stat. Comput. 10 (2000) 25–37) proposed a simple variational approach which relies on a family of tangent quadratic lower bounds of the logistic log-likelihood, thus restoring conjugacy between these approximate bounds and the Gaussian priors. This strategy is still implemented successfully, but few attempts have been made to formally understand the reasons underlying its excellent performance. Following a review on VB for logistic models, we cover this gap by providing a formal connection between the above bound and a recent Pólya-gamma data augmentation for logistic regression. Such a result places the computational methods associated with the aforementioned bounds within the framework of variational inference for conditionally conjugate exponential family models, thereby allowing recent advances for this class to be inherited also by the methods relying on Jaakkola and Jordan ( Stat. Comput. 10 (2000) 25–37).

Bradley Efron.

Source: Statistical Science, Volume 34, Number 2, 234--235.

Yixin Wang, Andrew C. Miller, David M. Blei.

Source: Statistical Science, Volume 34, Number 2, 229--233.