Federico Camerlenghi, David B. Dunson, Antonio Lijoi, Igor Prünster, Abel Rodríguez.

Source: Bayesian Analysis, Volume 14, Number 4, 1303--1356.

Abstract:
Discrete random structures are important tools in Bayesian nonparametrics and the resulting models have proven effective in density estimation, clustering, topic modeling and prediction, among others. In this paper, we consider nested processes and study the dependence structures they induce. Dependence ranges between homogeneity, corresponding to full exchangeability, and maximum heterogeneity, corresponding to (unconditional) independence across samples. The popular nested Dirichlet process is shown to degenerate to the fully exchangeable case when there are ties across samples at the observed or latent level. To overcome this drawback, inherent to nesting general discrete random measures, we introduce a novel class of latent nested processes. These are obtained by adding common and group-specific completely random measures and, then, normalizing to yield dependent random probability measures. We provide results on the partition distributions induced by latent nested processes, and develop a Markov Chain Monte Carlo sampler for Bayesian inferences. A test for distributional homogeneity across groups is obtained as a by-product. The results and their inferential implications are showcased on synthetic and real data.

Andrea Cremaschi, Raffaele Argiento, Katherine Shoemaker, Christine Peterson, Marina Vannucci.

Source: Bayesian Analysis, Volume 14, Number 4, 1271--1301.

Abstract:
Gaussian graphical models are useful tools for exploring network structures in multivariate normal data. In this paper we are interested in situations where data show departures from Gaussianity, therefore requiring alternative modeling distributions. The multivariate $t$ -distribution, obtained by dividing each component of the data vector by a gamma random variable, is a straightforward generalization to accommodate deviations from normality such as heavy tails. Since different groups of variables may be contaminated to a different extent, Finegold and Drton (2014) introduced the Dirichlet $t$ -distribution, where the divisors are clustered using a Dirichlet process. In this work, we consider a more general class of nonparametric distributions as the prior on the divisor terms, namely the class of normalized completely random measures (NormCRMs). To improve the effectiveness of the clustering, we propose modeling the dependence among the divisors through a nonparametric hierarchical structure, which allows for the sharing of parameters across the samples in the data set. This desirable feature enables us to cluster together different components of multivariate data in a parsimonious way. We demonstrate through simulations that this approach provides accurate graphical model inference, and apply it to a case study examining the dependence structure in radiomics data derived from The Cancer Imaging Atlas.

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.

Abhirup Datta, Sudipto Banerjee, James S. Hodges, Leiwen Gao.

Source: Bayesian Analysis, Volume 14, Number 4, 1221--1244.

Abstract:
Hierarchical models for regionally aggregated disease incidence data commonly involve region specific latent random effects that are modeled jointly as having a multivariate Gaussian distribution. The covariance or precision matrix incorporates the spatial dependence between the regions. Common choices for the precision matrix include the widely used ICAR model, which is singular, and its nonsingular extension which lacks interpretability. We propose a new parametric model for the precision matrix based on a directed acyclic graph (DAG) representation of the spatial dependence. Our model guarantees positive definiteness and, hence, in addition to being a valid prior for regional spatially correlated random effects, can also directly model the outcome from dependent data like images and networks. Theoretical results establish a link between the parameters in our model and the variance and covariances of the random effects. Simulation studies demonstrate that the improved interpretability of our model reaps benefits in terms of accurately recovering the latent spatial random effects as well as for inference on the spatial covariance parameters. Under modest spatial correlation, our model far outperforms the CAR models, while the performances are similar when the spatial correlation is strong. We also assess sensitivity to the choice of the ordering in the DAG construction using theoretical and empirical results which testify to the robustness of our model. We also present a large-scale public health application demonstrating the competitive performance of the model.

Lutz F. Gruber, Erica F. Stuber, Lyndsie S. Wszola, Joseph J. Fontaine.

Source: Bayesian Analysis, Volume 14, Number 4, 1173--1199.

Abstract:
We present an integrated open population model where the population dynamics are defined by a differential equation, and the related statistical model utilizes a Poisson binomial convolution likelihood. Key advantages of the proposed approach over existing open population models include the flexibility to predict related, but unobserved quantities such as total immigration or emigration over a specified time period, and more computationally efficient posterior simulation by elimination of the need to explicitly simulate latent immigration and emigration. The viability of the proposed method is shown in an in-depth analysis of outdoor recreation participation on public lands, where the surveyed populations changed rapidly and demographic population closure cannot be assumed even within a single day.

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).

Amir Bashir, Carlos M. Carvalho, P. Richard Hahn, M. Beatrix Jones.

Source: Bayesian Analysis, Volume 14, Number 4, 1075--1090.

Abstract:
A variety of computationally efficient Bayesian models for the covariance matrix of a multivariate Gaussian distribution are available. However, all produce a relatively dense estimate of the precision matrix, and are therefore unsatisfactory when one wishes to use the precision matrix to consider the conditional independence structure of the data. This paper considers the posterior predictive distribution of model fit for these covariance models. We then undertake post-processing of the Bayes point estimate for the precision matrix to produce a sparse model whose expected fit lies within the upper 95% of the posterior predictive distribution of fit. The impact of the method for selecting the zero elements of the precision matrix is evaluated. Good results were obtained using models that encouraged a sparse posterior (G-Wishart, Bayesian adaptive graphical lasso) and selection using credible intervals. We also find that this approach is easily extended to the problem of finding a sparse set of elements that differ across a set of precision matrices, a natural summary when a common set of variables is observed under multiple conditions. We illustrate our findings with moderate dimensional data examples from finance and metabolomics.

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.

Lizhen Lin, Niu Mu, Pokman Cheung, David Dunson.

Source: Bayesian Analysis, Volume 14, Number 3, 907--926.

Abstract:
Gaussian processes (GPs) are very widely used for modeling of unknown functions or surfaces in applications ranging from regression to classification to spatial processes. Although there is an increasingly vast literature on applications, methods, theory and algorithms related to GPs, the overwhelming majority of this literature focuses on the case in which the input domain corresponds to a Euclidean space. However, particularly in recent years with the increasing collection of complex data, it is commonly the case that the input domain does not have such a simple form. For example, it is common for the inputs to be restricted to a non-Euclidean manifold, a case which forms the motivation for this article. In particular, we propose a general extrinsic framework for GP modeling on manifolds, which relies on embedding of the manifold into a Euclidean space and then constructing extrinsic kernels for GPs on their images. These extrinsic Gaussian processes (eGPs) are used as prior distributions for unknown functions in Bayesian inferences. Our approach is simple and general, and we show that the eGPs inherit fine theoretical properties from GP models in Euclidean spaces. We consider applications of our models to regression and classification problems with predictors lying in a large class of manifolds, including spheres, planar shape spaces, a space of positive definite matrices, and Grassmannians. Our models can be readily used by practitioners in biological sciences for various regression and classification problems, such as disease diagnosis or detection. Our work is also likely to have impact in spatial statistics when spatial locations are on the sphere or other geometric spaces.

Mengyang Gu.

Source: Bayesian Analysis, Volume 14, Number 3, 877--905.

Abstract:
Gaussian stochastic process (GaSP) has been widely used in two fundamental problems in uncertainty quantification, namely the emulation and calibration of mathematical models. Some objective priors, such as the reference prior, are studied in the context of emulating (approximating) computationally expensive mathematical models. In this work, we introduce a new class of priors, called the jointly robust prior, for both the emulation and calibration. This prior is designed to maintain various advantages from the reference prior. In emulation, the jointly robust prior has an appropriate tail decay rate as the reference prior, and is computationally simpler than the reference prior in parameter estimation. Moreover, the marginal posterior mode estimation with the jointly robust prior can separate the influential and inert inputs in mathematical models, while the reference prior does not have this property. We establish the posterior propriety for a large class of priors in calibration, including the reference prior and jointly robust prior in general scenarios, but the jointly robust prior is preferred because the calibrated mathematical model typically predicts the reality well. The jointly robust prior is used as the default prior in two new R packages, called “RobustGaSP” and “RobustCalibration”, available on CRAN for emulation and calibration, respectively.

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.

Joseph Antonelli, Giovanni Parmigiani, Francesca Dominici.

Source: Bayesian Analysis, Volume 14, Number 3, 825--848.

Abstract:
In observational studies, estimation of a causal effect of a treatment on an outcome relies on proper adjustment for confounding. If the number of the potential confounders ( $p$ ) is larger than the number of observations ( $n$ ), then direct control for all potential confounders is infeasible. Existing approaches for dimension reduction and penalization are generally aimed at predicting the outcome, and are less suited for estimation of causal effects. Under standard penalization approaches (e.g. Lasso), if a variable $X_{j}$ is strongly associated with the treatment $T$ but weakly with the outcome $Y$ , the coefficient $eta_{j}$ will be shrunk towards zero thus leading to confounding bias. Under the assumption of a linear model for the outcome and sparsity, we propose continuous spike and slab priors on the regression coefficients $eta_{j}$ corresponding to the potential confounders $X_{j}$ . Specifically, we introduce a prior distribution that does not heavily shrink to zero the coefficients ( $eta_{j}$ s) of the $X_{j}$ s that are strongly associated with $T$ but weakly associated with $Y$ . We compare our proposed approach to several state of the art methods proposed in the literature. Our proposed approach has the following features: 1) it reduces confounding bias in high dimensional settings; 2) it shrinks towards zero coefficients of instrumental variables; and 3) it achieves good coverages even in small sample sizes. We apply our approach to the National Health and Nutrition Examination Survey (NHANES) data to estimate the causal effects of persistent pesticide exposure on triglyceride levels.

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.

L. F. South, A. N. Pettitt, C. C. Drovandi.

Source: Bayesian Analysis, Volume 14, Number 3, 773--796.

Abstract:
Sequential Monte Carlo (SMC) methods for sampling from the posterior of static Bayesian models are flexible, parallelisable and capable of handling complex targets. However, it is common practice to adopt a Markov chain Monte Carlo (MCMC) kernel with a multivariate normal random walk (RW) proposal in the move step, which can be both inefficient and detrimental for exploring challenging posterior distributions. We develop new SMC methods with independent proposals which allow recycling of all candidates generated in the SMC process and are embarrassingly parallelisable. A novel evidence estimator that is easily computed from the output of our independent SMC is proposed. Our independent proposals are constructed via flexible copula-type models calibrated with the population of SMC particles. We demonstrate through several examples that more precise estimates of posterior expectations and the marginal likelihood can be obtained using fewer likelihood evaluations than the more standard RW approach.

Julyan Arbel, Pierpaolo De Blasi, Igor Prünster.

Source: Bayesian Analysis, Volume 14, Number 3, 753--771.

Abstract:
In this paper we consider approximations to the popular Pitman–Yor process obtained by truncating the stick-breaking representation. The truncation is determined by a random stopping rule that achieves an almost sure control on the approximation error in total variation distance. We derive the asymptotic distribution of the random truncation point as the approximation error $epsilon$ goes to zero in terms of a polynomially tilted positive stable random variable. The practical usefulness and effectiveness of this theoretical result is demonstrated by devising a sampling algorithm to approximate functionals of the $epsilon$ -version of the Pitman–Yor process.

Stefano Peluso, Siddhartha Chib, Antonietta Mira.

Source: Bayesian Analysis, Volume 14, Number 3, 727--751.

Abstract:
We develop a general Bayesian semiparametric change-point model in which separate groups of structural parameters (for example, location and dispersion parameters) can each follow a separate multiple change-point process, driven by time-dependent transition matrices among the latent regimes. The distribution of the observations within regimes is unknown and given by a Dirichlet process mixture prior. The properties of the proposed model are studied theoretically through the analysis of inter-arrival times and of the number of change-points in a given time interval. The prior-posterior analysis by Markov chain Monte Carlo techniques is developed on a forward-backward algorithm for sampling the various regime indicators. Analysis with simulated data under various scenarios and an application to short-term interest rates are used to show the generality and usefulness of the proposed model.

Sohan Seth, Iain Murray, Christopher K. I. Williams.

Source: Bayesian Analysis, Volume 14, Number 3, 703--725.

Abstract:
Model criticism is usually carried out by assessing if replicated data generated under the fitted model looks similar to the observed data, see e.g. Gelman, Carlin, Stern, and Rubin (2004, p. 165). This paper presents a method for latent variable models by pulling back the data into the space of latent variables, and carrying out model criticism in that space. Making use of a model's structure enables a more direct assessment of the assumptions made in the prior and likelihood. We demonstrate the method with examples of model criticism in latent space applied to factor analysis, linear dynamical systems and Gaussian processes.

Yushu Shi, Michael Martens, Anjishnu Banerjee, Purushottam Laud.

Source: Bayesian Analysis, Volume 14, Number 3, 677--702.

Abstract:
Dirichlet process mixture (DPM) models provide flexible modeling for distributions of data as an infinite mixture of distributions from a chosen collection. Specifying priors for these models in individual data contexts can be challenging. In this paper, we introduce a scheme which requires the investigator to specify only simple scaling information. This is used to transform the data to a fixed scale on which a low information prior is constructed. Samples from the posterior with the rescaled data are transformed back for inference on the original scale. The low information prior is selected to provide a wide variety of components for the DPM to generate flexible distributions for the data on the fixed scale. The method can be applied to all DPM models with kernel functions closed under a suitable scaling transformation. Construction of the low information prior, however, is kernel dependent. Using DPM-of-Gaussians and DPM-of-Weibulls models as examples, we show that the method provides accurate estimates of a diverse collection of distributions that includes skewed, multimodal, and highly dispersed members. With the recommended priors, repeated data simulations show performance comparable to that of standard empirical estimates. Finally, we show weak convergence of posteriors with the proposed priors for both kernels considered.

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.

Marcos Oliveira Prates, Renato Martins Assunção, Erica Castilho Rodrigues.

Source: Bayesian Analysis, Volume 14, Number 2, 623--647.

Abstract:
Spatial confounding between the spatial random effects and fixed effects covariates has been recently discovered and showed that it may bring misleading interpretation to the model results. Techniques to alleviate this problem are based on decomposing the spatial random effect and fitting a restricted spatial regression. In this paper, we propose a different approach: a transformation of the geographic space to ensure that the unobserved spatial random effect added to the regression is orthogonal to the fixed effects covariates. Our approach, named SPOCK, has the additional benefit of providing a fast and simple computational method to estimate the parameters. Also, it does not constrain the distribution class assumed for the spatial error term. A simulation study and real data analyses are presented to better understand the advantages of the new method in comparison with the existing ones.

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.

Łukasz Rajkowski.

Source: Bayesian Analysis, Volume 14, Number 2, 477--494.

Abstract:
Mixture models are a natural choice in many applications, but it can be difficult to place an a priori upper bound on the number of components. To circumvent this, investigators are turning increasingly to Dirichlet process mixture models (DPMMs). It is therefore important to develop an understanding of the strengths and weaknesses of this approach. This work considers the MAP (maximum a posteriori) clustering for the Gaussian DPMM (where the cluster means have Gaussian distribution and, for each cluster, the observations within the cluster have Gaussian distribution). Some desirable properties of the MAP partition are proved: ‘almost disjointness’ of the convex hulls of clusters (they may have at most one point in common) and (with natural assumptions) the comparability of sizes of those clusters that intersect any fixed ball with the number of observations (as the latter goes to infinity). Consequently, the number of such clusters remains bounded. Furthermore, if the data arises from independent identically distributed sampling from a given distribution with bounded support then the asymptotic MAP partition of the observation space maximises a function which has a straightforward expression, which depends only on the within-group covariance parameter. As the operator norm of this covariance parameter decreases, the number of clusters in the MAP partition becomes arbitrarily large, which may lead to the overestimation of the number of mixture components.

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.

M. W. McLean, M. P. Wand.

Source: Bayesian Analysis, Volume 14, Number 2, 371--398.

Abstract:
We build on recent work concerning message passing approaches to approximate fitting and inference for arbitrarily large regression models. The focus is on regression models where the response variable is modeled to have an elaborate distribution, which is loosely defined to mean a distribution that is more complicated than common distributions such as those in the Bernoulli, Poisson and Normal families. Examples of elaborate response families considered here are the Negative Binomial and $t$ families. Variational message passing is more challenging due to some of the conjugate exponential families being non-standard and numerical integration being needed. Nevertheless, a factor graph fragment approach means the requisite calculations only need to be done once for a particular elaborate response distribution family. Computer code can be compartmentalized, including that involving numerical integration. A major finding of this work is that the modularity of variational message passing extends to elaborate response regression models.

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).

Lloyd T. Elliott, Maria De Iorio, Stefano Favaro, Kaustubh Adhikari, Yee Whye Teh.

Source: Bayesian Analysis, Volume 14, Number 2, 313--339.

Abstract:
We propose a Bayesian nonparametric model to infer population admixture, extending the hierarchical Dirichlet process to allow for correlation between loci due to linkage disequilibrium. Given multilocus genotype data from a sample of individuals, the proposed model allows inferring and classifying individuals as unadmixed or admixed, inferring the number of subpopulations ancestral to an admixed population and the population of origin of chromosomal regions. Our model does not assume any specific mutation process, and can be applied to most of the commonly used genetic markers. We present a Markov chain Monte Carlo (MCMC) algorithm to perform posterior inference from the model and we discuss some methods to summarize the MCMC output for the analysis of population admixture. Finally, we demonstrate the performance of the proposed model in a real application, using genetic data from the ectodysplasin-A receptor (EDAR) gene, which is considered to be ancestry-informative due to well-known variations in allele frequency as well as phenotypic effects across ancestry. The structure analysis of this dataset leads to the identification of a rare haplotype in Europeans. We also conduct a simulated experiment and show that our algorithm outperforms parametric methods.

Peter D. Hoff

Source: Bayesian Anal., Volume 6, Number 2, 179--196.

Abstract:
Modern datasets are often in the form of matrices or arrays, potentially having correlations along each set of data indices. For example, data involving repeated measurements of several variables over time may exhibit temporal correlation as well as correlation among the variables. A possible model for matrix-valued data is the class of matrix normal distributions, which is parametrized by two covariance matrices, one for each index set of the data. In this article we discuss an extension of the matrix normal model to accommodate multidimensional data arrays, or tensors. We show how a particular array-matrix product can be used to generate the class of array normal distributions having separable covariance structure. We derive some properties of these covariance structures and the corresponding array normal distributions, and show how the array-matrix product can be used to define a semi-conjugate prior distribution and calculate the corresponding posterior distribution. We illustrate the methodology in an analysis of multivariate longitudinal network data which take the form of a four-way array.

Ruosi Guo, Chunming Zhang, Zhengjun Zhang.

Source: Statistical Science, Volume 35, Number 1, 145--157.

Abstract:
In many scientific disciplines, finding hidden influential factors behind observational data is essential but challenging. The majority of existing approaches, such as the independent component analysis (${mathrm{ICA}}$), rely on linear transformation, that is, true signals are linear combinations of hidden components. Motivated from analyzing nonlinear temporal signals in neuroscience, genetics, and finance, this paper proposes the “maximum independent component analysis” (${mathrm{MaxICA}}$), based on max-linear combinations of components. In contrast to existing methods, ${mathrm{MaxICA}}$ benefits from focusing on significant major components while filtering out ignorable components. A major tool for parameter learning of ${mathrm{MaxICA}}$ is an augmented genetic algorithm, consisting of three schemes for the elite weighted sum selection, randomly combined crossover, and dynamic mutation. Extensive empirical evaluations demonstrate the effectiveness of ${mathrm{MaxICA}}$ in either extracting max-linearly combined essential sources in many applications or supplying a better approximation for nonlinearly combined source signals, such as $mathrm{EEG}$ recordings analyzed in this paper.

Khanh N. Dinh, Roman Jaksik, Marek Kimmel, Amaury Lambert, Simon Tavaré.

Source: Statistical Science, Volume 35, Number 1, 129--144.

Abstract:
Recent years have seen considerable work on inference about cancer evolution from mutations identified in cancer samples. Much of the modeling work has been based on classical models of population genetics, generalized to accommodate time-varying cell population size. Reverse-time, genealogical views of such models, commonly known as coalescents, have been used to infer aspects of the past of growing populations. Another approach is to use branching processes, the simplest scenario being the classical linear birth-death process. Inference from evolutionary models of DNA often exploits summary statistics of the sequence data, a common one being the so-called Site Frequency Spectrum (SFS). In a bulk tumor sequencing experiment, we can estimate for each site at which a novel somatic point mutation has arisen, the proportion of cells that carry that mutation. These numbers are then grouped into collections of sites which have similar mutant fractions. We examine how the SFS based on birth-death processes differs from those based on the coalescent model. This may stem from the different sampling mechanisms in the two approaches. However, we also show that despite this, they are quantitatively comparable for the range of parameters typical for tumor cell populations. We also present a model of tumor evolution with selective sweeps, and demonstrate how it may help in understanding the history of a tumor as well as the influence of data pre-processing. We illustrate the theory with applications to several examples from The Cancer Genome Atlas tumors.

Divyansh Agarwal, Jingshu Wang, Nancy R. Zhang.

Source: Statistical Science, Volume 35, Number 1, 112--128.

Abstract:
Single cell sequencing technologies are transforming biomedical research. However, due to the inherent nature of the data, single cell RNA sequencing analysis poses new computational and statistical challenges. We begin with a survey of a selection of topics in this field, with a gentle introduction to the biology and a more detailed exploration of the technical noise. We consider in detail the problem of single cell data denoising, sometimes referred to as “imputation” in the relevant literature. We discuss why this is not a typical statistical imputation problem, and review current approaches to this problem. We then explore why the use of denoised values in downstream analyses invites novel statistical insights, and how denoising uncertainty should be accounted for to yield valid statistical inference. The utilization of denoised or imputed matrices in statistical inference is not unique to single cell genomics, and arises in many other fields. We describe the challenges in this type of analysis, discuss some preliminary solutions, and highlight unresolved issues.

Thomas Staudt, Timo Aspelmeier, Oskar Laitenberger, Claudia Geisler, Alexander Egner, Axel Munk.

Source: Statistical Science, Volume 35, Number 1, 92--111.

Abstract:
Super-resolution microscopy is rapidly gaining importance as an analytical tool in the life sciences. A compelling feature is the ability to label biological units of interest with fluorescent markers in (living) cells and to observe them with considerably higher resolution than conventional microscopy permits. The images obtained this way, however, lack an absolute intensity scale in terms of numbers of fluorophores observed. In this article, we discuss state of the art methods to count such fluorophores and statistical challenges that come along with it. In particular, we suggest a modeling scheme for time series generated by single-marker-switching (SMS) microscopy that makes it possible to quantify the number of markers in a statistically meaningful manner from the raw data. To this end, we model the entire process of photon generation in the fluorophore, their passage through the microscope, detection and photoelectron amplification in the camera, and extraction of time series from the microscopic images. At the heart of these modeling steps is a careful description of the fluorophore dynamics by a novel hidden Markov model that operates on two timescales (HTMM). Besides the fluorophore number, information about the kinetic transition rates of the fluorophore’s internal states is also inferred during estimation. We comment on computational issues that arise when applying our model to simulated or measured fluorescence traces and illustrate our methodology on simulated data.

Chao Du, S. C. Kou.

Source: Statistical Science, Volume 35, Number 1, 75--91.

Abstract:
Toward the last quarter of the 20th century, the emergence of single-molecule experiments enabled scientists to track and study individual molecules’ dynamic properties in real time. Unlike macroscopic systems’ dynamics, those of single molecules can only be properly described by stochastic models even in the absence of external noise. Consequently, statistical methods have played a key role in extracting hidden information about molecular dynamics from data obtained through single-molecule experiments. In this article, we survey the major statistical methodologies used to analyze single-molecule experimental data. Our discussion is organized according to the types of stochastic models used to describe single-molecule systems as well as major experimental data collection techniques. We also highlight challenges and future directions in the application of statistical methodologies to single-molecule experiments.

Yazhen Wang, Xinyu Song.

Source: Statistical Science, Volume 35, Number 1, 51--74.

Abstract:
Quantum science and quantum technology are of great current interest in multiple frontiers of many scientific fields ranging from computer science to physics and chemistry, and from engineering to mathematics and statistics. Their developments will likely lead to a new wave of scientific revolutions and technological innovations in a wide range of scientific studies and applications. This paper provides a brief review on quantum communication, quantum information, quantum computation, quantum simulation, and quantum metrology. We present essential quantum properties, illustrate relevant concepts of quantum science and quantum technology, and discuss their scientific developments. We point out the need for statistical analysis in their developments, as well as their potential applications to and impacts on statistics and data science.

Peter J. Diggle, Emanuele Giorgi, Julienne Atsame, Sylvie Ntsame Ella, Kisito Ogoussan, Katherine Gass.

Source: Statistical Science, Volume 35, Number 1, 42--50.

Abstract:
Vector-borne diseases have long presented major challenges to the health of rural communities in the wet tropical regions of the world, but especially in sub-Saharan Africa. In this paper, we describe the contribution that statistical modelling has made to the global elimination programme for one vector-borne disease, onchocerciasis. We explain why information on the spatial distribution of a second vector-borne disease, Loa loa, is needed before communities at high risk of onchocerciasis can be treated safely with mass distribution of ivermectin, an antifiarial medication. We show how a model-based geostatistical analysis of Loa loa prevalence survey data can be used to map the predictive probability that each location in the region of interest meets a WHO policy guideline for safe mass distribution of ivermectin and describe two applications: one is to data from Cameroon that assesses prevalence using traditional blood-smear microscopy; the other is to Africa-wide data that uses a low-cost questionnaire-based method. We describe how a recent technological development in image-based microscopy has resulted in a change of emphasis from prevalence alone to the bivariate spatial distribution of prevalence and the intensity of infection among infected individuals. We discuss how statistical modelling of the kind described here can contribute to health policy guidelines and decision-making in two ways. One is to ensure that, in a resource-limited setting, prevalence surveys are designed, and the resulting data analysed, as efficiently as possible. The other is to provide an honest quantification of the uncertainty attached to any binary decision by reporting predictive probabilities that a policy-defined condition for action is or is not met.

Michael L. Stein.

Source: Statistical Science, Volume 35, Number 1, 31--41.

Abstract:
Climate science is a field that is arguably both data-rich and data-poor. Data rich in that huge and quickly increasing amounts of data about the state of the climate are collected every day. Data poor in that important aspects of the climate are still undersampled, such as the deep oceans and some characteristics of the upper atmosphere. Data rich in that modern climate models can produce climatological quantities over long time periods with global coverage, including quantities that are difficult to measure and under conditions for which there is no data presently. Data poor in that the correspondence between climate model output to the actual climate, especially for future climate change due to human activities, is difficult to assess. The scope for fruitful interactions between climate scientists and statisticians is great, but requires serious commitments from researchers in both disciplines to understand the scientific and statistical nuances arising from the complex relationships between the data and the real-world problems. This paper describes a small fraction of some of the intellectual challenges that occur at the interface between climate science and statistics, including inferences for extremes for processes with seasonality and long-term trends, the use of climate model ensembles for studying extremes, the scope for using new data sources for studying space-time characteristics of environmental processes and a discussion of non-Gaussian space-time process models for climate variables. The paper concludes with a call to the statistical community to become more engaged in one of the great scientific and policy issues of our time, anthropogenic climate change and its impacts.

Adam R. Brentnall, Jack Cuzick.

Source: Statistical Science, Volume 35, Number 1, 14--30.

Abstract:
Strategies to prevent cancer and diagnose it early when it is most treatable are needed to reduce the public health burden from rising disease incidence. Risk assessment is playing an increasingly important role in targeting individuals in need of such interventions. For breast cancer many individual risk factors have been well understood for a long time, but the development of a fully comprehensive risk model has not been straightforward, in part because there have been limited data where joint effects of an extensive set of risk factors may be estimated with precision. In this article we first review the approach taken to develop the IBIS (Tyrer–Cuzick) model, and describe recent updates. We then review and develop methods to assess calibration of models such as this one, where the risk of disease allowing for competing mortality over a long follow-up time or lifetime is estimated. The breast cancer risk model model and calibration assessment methods are demonstrated using a cohort of 132,139 women attending mammography screening in the State of Washington, USA.

Zhixiang Lin, Mahdi Zamanighomi, Timothy Daley, Shining Ma, Wing Hung Wong.

Source: Statistical Science, Volume 35, Number 1, 2--13.

Abstract:
Unsupervised methods, including clustering methods, are essential to the analysis of single-cell genomic data. Model-based clustering methods are under-explored in the area of single-cell genomics, and have the advantage of quantifying the uncertainty of the clustering result. Here we develop a model-based approach for the integrative analysis of single-cell chromatin accessibility and gene expression data. We show that combining these two types of data, we can achieve a better separation of the underlying cell types. An efficient Markov chain Monte Carlo algorithm is also developed.

Source: Statistical Science, Volume 35, Number 1, 1--1.

Junhui Cai, Avishai Mandelbaum, Chaitra H. Nagaraja, Haipeng Shen, Linda Zhao.

Source: Statistical Science, Volume 34, Number 4, 669--691.

Abstract:
Statisticians are finding their place in the emerging field of data science. However, many issues considered “new” in data science have long histories in statistics. Examples of using statistical thinking are illustrated, which range from exploratory data analysis to measuring uncertainty to accommodating nonrandom samples. These examples are then applied to service networks, baseball predictions and official statistics.