Markov chain Monte Carlo is a key computational tool in Bayesian statistics, but it can be challenging to monitor the convergence of an iterative stochastic algorithm. In this paper we show that the convergence diagnostic $\hat{R}$ of Gelman and Rubin (1992) has serious flaws. Traditional $\hat{R}$ will fail to correctly diagnose convergence failures when the chain has a heavy tail or when the variance varies across the chains. In this paper we propose an alternative rank-based diagnostic that fixes these problems. We also introduce a collection of quantile-based local efficiency measures, along with a practical approach for computing Monte Carlo error estimates for quantiles. We suggest that common trace plots should be replaced with rank plots from multiple chains. Finally, we give recommendations for how these methods should be used in practice.

A central theme in the field of survey statistics is estimating population-level quantities through data coming from potentially non-representative samples of the population. Multilevel regression and poststratification (MRP), a model-based approach, is gaining traction against the traditional weighted approach for survey estimates. MRP estimates are susceptible to bias if there is an underlying structure that the methodology does not capture. This work aims to provide a new framework for specifying structured prior distributions that lead to bias reduction in MRP estimates. We use simulation studies to explore the benefit of these prior distributions and demonstrate their efficacy on non-representative US survey data. We show that structured prior distributions offer absolute bias reduction and variance reduction for posterior MRP estimates in a large variety of data regimes.

Sequential Monte Carlo (SMC) samplers are an alternative to MCMC for Bayesian computation. However, their performance depends strongly on the Markov kernels used to rejuvenate particles. We discuss how to calibrate automatically (using the current particles) Hamiltonian Monte Carlo kernels within SMC. To do so, we build upon the adaptive SMC approach of Fearnhead and Taylor (2013), and we also suggest alternative methods. We illustrate the advantages of using HMC kernels within an SMC sampler via an extensive numerical study.

Bayesian experimental design (BED) is a framework that uses statistical models and decision making under uncertainty to optimise the cost and performance of a scientific experiment. Sequential BED, as opposed to static BED, considers the scenario where we can sequentially update our beliefs about the model parameters through data gathered in the experiment. A class of models of particular interest for the natural and medical sciences are implicit models, where the data generating distribution is intractable, but sampling from it is possible. Even though there has been a lot of work on static BED for implicit models in the past few years, the notoriously difficult problem of sequential BED for implicit models has barely been touched upon. We address this gap in the literature by devising a novel sequential design framework for parameter estimation that uses the Mutual Information (MI) between model parameters and simulated data as a utility function to find optimal experimental designs, which has not been done before for implicit models. Our approach uses likelihood-free inference by ratio estimation to simultaneously estimate posterior distributions and the MI. During the sequential BED procedure we utilise Bayesian optimisation to help us optimise the MI utility. We find that our framework is efficient for the various implicit models tested, yielding accurate parameter estimates after only a few iterations.

Network meta-analysis has gained popularity in the last decade as a method for comparing the efficacy/safety of multiple medical interventions by synthesizing data across clinical studies. Bayesian methods for network meta-analysis have undergone further development than frequentist methods and are more convenient to use. Most of the current literature pertains to connected networks but disconnected networks commonly arise. There is not at the moment a trusted gold-standard approach to analyze disconnected networks. Intuitively, the standard method for analyzing connected networks, which is contrast-based, does not seem useful in disconnected networks, but this has not been explained rigorously. Our work is the first to provide the theoretical groundwork for understanding how evidence flows within Bayesian contrast-based models of disconnected networks. We achieve this by quantifying the ratio of posterior to prior variance of disconnected treatment contrasts. We show that when using an uninformative prior on the treatment contrasts, the standard approach is not useful to analyze disconnected networks (even when the number of studies, treatments or patients is large); however, it can be useful under moderately informative priors, which can be informed from additional observational data when available. A simulation study provides a demonstration of the theoretical results and explores non-asymptotic cases. An illustration on a real-world dataset is provided.

This paper proposes randomized controlled clinical trial design to evaluate external cooling as a means to control fever and thereby reduce mortality in patients with septic shock. The trial will include concurrent external cooling and control arms while adaptively incorporating historical control arm data. Bayesian group sequential monitoring will be done using a posterior comparative test based on the 60-day survival distribution in each concurrent arm. Posterior inference will follow from a Bayesian discrete time survival model that facilitates adaptive incorporation of the historical control data through an innovative regression framework with a multivariate spike-and-slab prior distribution on the historical bias parameters. For each interim test, the amount of information borrowed from the historical control data will be determined adaptively in a manner that reflects the degree of agreement between historical and concurrent control arm data. Guidance is provided for selecting Bayesian posterior probability group-sequential monitoring boundaries. Simulation results elucidating how the proposed method borrows strength from the historical control data are reported. In the absence of historical control arm bias, the proposed design controls the type I error rate and provides substantially larger power than reasonable comparators, whereas in the presence bias of varying magnitude, type I error rate inflation is curbed.

We provide a nonparametric spectral approach to the modeling of correlation functions on spheres. The sequence of Schoenberg coefficients and their associated covariance functions are treated as random rather than assuming a parametric form. We propose a stick-breaking representation for the spectrum, and show that such a choice spans the support of the class of geodesically isotropic covariance functions under uniform convergence. Further, we examine the first order properties of such representation, from which geometric properties can be inferred, in terms of Hölder continuity, of the associated Gaussian random field. The properties of the posterior, in terms of existence, uniqueness, and Lipschitz continuity, are then inspected. Our findings are validated with MCMC simulations and illustrated using a global data set on surface temperatures.

We propose the use of a score based working likelihood function for quantile regression which can perform inference for multiple conditional quantiles of an arbitrary number. We show that the proposed likelihood can be used in a Bayesian framework leading to valid frequentist inference, whereas the commonly used asymmetric Laplace working likelihood leads to invalid interval estimations and requires further correction. For computation, we propose a novel adaptive importance sampling algorithm to compute important posterior summaries such as the posterior mean and the covariance matrix. Our proposed approach makes it feasible to perform valid inference for parameters such as the slope differences at different quantile levels, which is either not possible or cumbersome using existing Bayesian approaches. Empirical results demonstrate that the proposed likelihood has good estimation and inferential properties and that the proposed computational algorithm is more efficient than its competitors.

Bayesian inference for models with intractable likelihood functions represents a challenging suite of problems in modern statistics. In this work we analyse the Conway-Maxwell-Poisson (COM-Poisson) distribution, a two parameter generalisation of the Poisson distribution. COM-Poisson regression modelling allows the flexibility to model dispersed count data as part of a generalised linear model (GLM) with a COM-Poisson response, where exogenous covariates control the mean and dispersion level of the response. The major difficulty with COM-Poisson regression is that the likelihood function contains multiple intractable normalising constants and is not amenable to standard inference and Markov Chain Monte Carlo (MCMC) techniques. Recent work by Chanialidis et al. (2018) has seen the development of a sampler to draw random variates from the COM-Poisson likelihood using a rejection sampling algorithm. We provide a new rejection sampler for the COM-Poisson distribution which significantly reduces the central processing unit (CPU) time required to perform inference for COM-Poisson regression models. An extension of this work shows that for any intractable likelihood function with an associated rejection sampler it is possible to construct unbiased estimators of the intractable likelihood which proves useful for model selection or for use within pseudo-marginal MCMC algorithms (Andrieu and Roberts, 2009). We demonstrate all of these methods on a real-world dataset of takeover bids.

We consider exact algorithms for Bayesian inference with model selection priors (including spike-and-slab priors) in the sparse normal sequence model. Because the best existing exact algorithm becomes numerically unstable for sample sizes over $n=500$, there has been much attention for alternative approaches like approximate algorithms (Gibbs sampling, variational Bayes, etc.), shrinkage priors (e.g. the Horseshoe prior and the Spike-and-Slab LASSO) or empirical Bayesian methods. However, by introducing algorithmic ideas from online sequential prediction, we show that exact calculations are feasible for much larger sample sizes: for general model selection priors we reach $n=25000$, and for certain spike-and-slab priors we can easily reach $n=100000$. We further prove a de Finetti-like result for finite sample sizes that characterizes exactly which model selection priors can be expressed as spike-and-slab priors. The computational speed and numerical accuracy of the proposed methods are demonstrated in experiments on simulated data, on a differential gene expression data set, and to compare the effect of multiple hyper-parameter settings in the beta-binomial prior. In our experimental evaluation we compute guaranteed bounds on the numerical accuracy of all new algorithms, which shows that the proposed methods are numerically reliable whereas an alternative based on long division is not.

It is often claimed that Bayesian methods, in particular Bayes factor methods for hypothesis testing, can deal with optional stopping. We first give an overview, using elementary probability theory, of three different mathematical meanings that various authors give to this claim: (1) stopping rule independence, (2) posterior calibration and (3) (semi-) frequentist robustness to optional stopping. We then prove theorems to the effect that these claims do indeed hold in a general measure-theoretic setting. For claims of type (2) and (3), such results are new. By allowing for non-integrable measures based on improper priors, we obtain particularly strong results for the practically important case of models with nuisance parameters satisfying a group invariance (such as location or scale). We also discuss the practical relevance of (1)–(3), and conclude that whether Bayes factor methods actually perform well under optional stopping crucially depends on details of models, priors and the goal of the analysis.

Fitted probabilities from widely used Bayesian multinomial probit models can depend strongly on the choice of a base category, which is used to uniquely identify the parameters of the model. This paper proposes a novel identification strategy, and associated prior distribution for the model parameters, that renders the prior symmetric with respect to relabeling the outcome categories. The new prior permits an efficient Gibbs algorithm that samples rank-deficient covariance matrices without resorting to Metropolis-Hastings updates.

In this paper we describe Bayesian inferential methods for data modeled by controlled branching processes that account for model robustness via the use of disparities. Under regularity conditions, we establish that estimators obtained using disparity-based posterior, such as expected and maximum a posteriori estimates, are consistent and efficient under the posited model. Additionally, we establish that the estimates are robust to model misspecification and presence of outliers. To this end, we develop several fundamental ideas relating minimum disparity estimators to Bayesian estimators obtained using the disparity-based posterior, for dependent tree-structured data. We illustrate the methodology through a simulated example and apply our methods to a real data set from cell kinetics.

Estimation of correlation matrices is a challenging problem due to the notorious positive-definiteness constraint and high-dimensionality. Reparameterizing Cholesky factors of correlation matrices in terms of angles or hyperspherical coordinates where the angles vary freely in the range $[0,\pi )$ has become popular in the last two decades. However, it has not been used in Bayesian estimation of correlation matrices perhaps due to lack of clear statistical relevance and suitable priors for the angles. In this paper, we show for the first time that for longitudinal data these angles are the inverse cosine of the semi-partial correlations (SPCs). This simple connection makes it possible to introduce physically meaningful selection and shrinkage priors on the angles or correlation matrices with emphasis on selection (sparsity) and shrinking towards longitudinal structure. Our method deals effectively with the positive-definiteness constraint in posterior computation. We compare the performance of our Bayesian estimation based on angles with some recent methods based on partial autocorrelations through simulation and apply the method to a data related to clinical trial on smoking.

We study the convergence properties of the Gibbs Sampler in the context of posterior distributions arising from Bayesian analysis of conditionally Gaussian hierarchical models. We develop a multigrid approach to derive analytic expressions for the convergence rates of the algorithm for various widely used model structures, including nested and crossed random effects. Our results apply to multilevel models with an arbitrary number of layers in the hierarchy, while most previous work was limited to the two-level nested case. The theoretical results provide explicit and easy-to-implement guidelines to optimize practical implementations of the Gibbs Sampler, such as indications on which parametrization to choose (e.g. centred and non-centred), which constraint to impose to guarantee statistical identifiability, and which parameters to monitor in the diagnostic process. Simulations suggest that the results are informative also in the context of non-Gaussian distributions and more general MCMC schemes, such as gradient-based ones.

We consider the problem of parametric statistical inference when likelihood computations are prohibitively expensive but sampling from the model is possible. Several so-called likelihood-free methods have been developed to perform inference in the absence of a likelihood function. The popular synthetic likelihood approach infers the parameters by modelling summary statistics of the data by a Gaussian probability distribution. In another popular approach called approximate Bayesian computation, the inference is performed by identifying parameter values for which the summary statistics of the simulated data are close to those of the observed data. Synthetic likelihood is easier to use as no measure of “closeness” is required but the Gaussianity assumption is often limiting. Moreover, both approaches require judiciously chosen summary statistics. We here present an alternative inference approach that is as easy to use as synthetic likelihood but not as restricted in its assumptions, and that, in a natural way, enables automatic selection of relevant summary statistic from a large set of candidates. The basic idea is to frame the problem of estimating the posterior as a problem of estimating the ratio between the data generating distribution and the marginal distribution. This problem can be solved by logistic regression, and including regularising penalty terms enables automatic selection of the summary statistics relevant to the inference task. We illustrate the general theory on canonical examples and employ it to perform inference for challenging stochastic nonlinear dynamical systems and high-dimensional summary statistics.

Two key challenges in modern statistical applications are the large amount of information recorded per individual, and that such data are often not collected all at once but in batches. These batch effects can be complex, causing distortions in both mean and variance. We propose a novel sparse latent factor regression model to integrate such heterogeneous data. The model provides a tool for data exploration via dimensionality reduction and sparse low-rank covariance estimation while correcting for a range of batch effects. We study the use of several sparse priors (local and non-local) to learn the dimension of the latent factors. We provide a flexible methodology for sparse factor regression which is not limited to data with batch effects. Our model is fitted in a deterministic fashion by means of an EM algorithm for which we derive closed-form updates, contributing a novel scalable algorithm for non-local priors of interest beyond the immediate scope of this paper. We present several examples, with a focus on bioinformatics applications. Our results show an increase in the accuracy of the dimensionality reduction, with non-local priors substantially improving the reconstruction of factor cardinality. The results of our analyses illustrate how failing to properly account for batch effects can result in unreliable inference. Our model provides a novel approach to latent factor regression that balances sparsity with sensitivity in scenarios both with and without batch effects and is highly computationally efficient.

A Bayesian model for individual insurance claims is proposed which accounts for the multivariate multilevel features of the claims, including multiple claimants for the same event, each of whom may receive benefits under different coverages. A Bayesian approach makes it possible to account for missing values in the covariates and partial information contained in open files, thereby avoiding sampling bias induced when unsettled claims are ignored. For a given claim, the combination of coverages under which payments are made is modeled as a type with multinomial regression. The presence of legal and expert fees follows a logistic regression given the type. The non-zero claim amounts are then modeled with log skewed normal regressions linked by a Student $t$ copula. The Bayesian framework yields a predictive distribution for the amounts paid, including parameter risk and process risk, while handling missing covariates and open files. The approach is illustrated with Accident Benefits car insurance claims from a Canadian company.

Data quality from poor and socially deprived regions have given rise to many statistical challenges. One of them is the underreporting of vital events leading to biased estimates for the associated risks. To deal with underreported count data, models based on compound Poisson distributions have been commonly assumed. To be identifiable, such models usually require extra and strong information about the probability of reporting the event in all areas of interest, which is not always available. We introduce a novel approach for the compound Poisson model assuming that the areas are clustered according to their data quality. We leverage these clusters to create a hierarchical structure in which the reporting probabilities decrease as we move from the best group to the worst ones. We obtain constraints for model identifiability and prove that only prior information about the reporting probability in areas experiencing the best data quality is required. Several approaches to model the uncertainty about the reporting probabilities are presented, including reference priors. Different features regarding the proposed methodology are studied through simulation. We apply our model to map the early neonatal mortality risks in Minas Gerais, a Brazilian state that presents heterogeneous characteristics and a relevant socio-economical inequality.

We discuss Bayesian model uncertainty analysis and forecasting in sequential dynamic modeling of multivariate time series. The perspective is that of a decision-maker with a specific forecasting objective that guides thinking about relevant models. Based on formal Bayesian decision-theoretic reasoning, we develop a time-adaptive approach to exploring, weighting, combining and selecting models that differ in terms of predictive variables included. The adaptivity allows for changes in the sets of favored models over time, and is guided by the specific forecasting goals. A synthetic example illustrates how decision-guided variable selection differs from traditional Bayesian model uncertainty analysis and standard model averaging. An applied study in one motivating application of long-term macroeconomic forecasting highlights the utility of the new approach in terms of improving predictions as well as its ability to identify and interpret different sets of relevant models over time with respect to specific, defined forecasting goals.

Consider the situation where an analyst has a Bayesian statistical model that performs well for continuous data. However, suppose the observed dataset consists of multiple response-types (e.g., continuous, count-valued, Bernoulli trials, etc.), which are distributed from more than one class of distributions. We refer to these types of data as “multiple response-type” datasets. The goal of this article is to introduce a reasonable easy-to-implement all-purpose method that “converts” a Bayesian statistical model for continuous responses (call this the preferred model) into a Bayesian model for multiple response-type datasets. To do this, we consider a transformation of the multiple response-type data, such that the transformed data can be reasonably modeled using the preferred model. What is unique with our strategy is that we treat the transformations as unknown and use a Bayesian approach to model this uncertainty. The implementation of our Bayesian approach to unknown transformations is straightforward, and involves two steps. The first step produces posterior replicates of the transformed multiple response-type data from a latent conjugate multivariate (LCM) model. The second step involves generating values from the posterior distribution implied by the preferred model. We demonstrate the flexibility of our model through an application to Bayesian additive regression trees (BART) and a spatio-temporal mixed effects (SME) model. We provide a thorough joint multiple response-type spatio-temporal analysis of coronavirus disease 2019(COVID-19) cases, the adjusted closing price of the Dow Jones Industrial (DJI), and Google Trends data.

Approximate Bayesian computation (ABC) is a simulation-based likelihood-free method applicable to both model selection and parameter estimation. ABC parameter estimation requires the ability to forward simulate datasets from a candidate model, but because the sizes of the observed and simulated datasets usually need to match, this can be computationally expensive. Additionally, since ABC inference is based on comparisons of summary statistics computed on the observed and simulated data, using computationally expensive summary statistics can lead to further losses in efficiency. ABC has recently been applied to the family of mechanistic network models, an area that has traditionally lacked tools for inference and model choice. Mechanistic models of network growth repeatedly add nodes to a network until it reaches the size of the observed network, which may be of the order of millions of nodes. With ABC, this process can quickly become computationally prohibitive due to the resource intensive nature of network simulations and evaluation of summary statistics. We propose two methodological developments to enable the use of ABC for inference in models for large growing networks. First, to save time needed for forward simulating model realizations, we propose a procedure to extrapolate (via both least squares and Gaussian processes) summary statistics from small to large networks. Second, to reduce computation time for evaluating summary statistics, we use sample-based rather than census-based summary statistics. We show that the ABC posterior obtained through this approach, which adds two additional layers of approximation to the standard ABC, is similar to a classic ABC posterior. Although we deal with growing network models, both extrapolated summaries and sampled summaries are expected to be relevant in other ABC settings where the data are generated incrementally.

With larger data at their disposal, scientists are emboldened to tackle complex questions that require sophisticated statistical models. It is not unusual for the latter to have likelihood functions that elude analytical formulations. Even under such adversity, when one can simulate from the sampling distribution, Bayesian analysis can be conducted using approximate methods such as Approximate Bayesian Computation (ABC) or Bayesian Synthetic Likelihood (BSL). A significant drawback of these methods is that the number of required simulations can be prohibitively large, thus severely limiting their scope. In this paper we design perturbed MCMC samplers that can be used within the ABC and BSL paradigms to significantly accelerate computation while maintaining control on computational efficiency. The proposed strategy relies on recycling samples from the chain’s past. The algorithmic design is supported by a theoretical analysis while practical performance is examined via a series of simulation examples and data analyses.

Particle Markov chain Monte Carlo (pMCMC) is now a popular method for performing Bayesian statistical inference on challenging state space models (SSMs) with unknown static parameters. It uses a particle filter (PF) at each iteration of an MCMC algorithm to unbiasedly estimate the likelihood for a given static parameter value. However, pMCMC can be computationally intensive when a large number of particles in the PF is required, such as when the data are highly informative, the model is misspecified and/or the time series is long. In this paper we exploit the ensemble Kalman filter (EnKF) developed in the data assimilation literature to speed up pMCMC. We replace the unbiased PF likelihood with the biased EnKF likelihood estimate within MCMC to sample over the space of the static parameter. On a wide class of different non-linear SSM models, we demonstrate that our extended ensemble MCMC (eMCMC) methods can significantly reduce the computational cost whilst maintaining reasonable accuracy. We also propose several extensions of the vanilla eMCMC algorithm to further improve computational efficiency. Computer code to implement our methods on all the examples can be downloaded from https://github.com/cdrovandi/Ensemble-MCMC.

The median probability model (MPM) (Barbieri and Berger, 2004) is defined as the model consisting of those variables whose marginal posterior probability of inclusion is at least $0.5$. The MPM rule yields the best single model for prediction in orthogonal and nested correlated designs. This result was originally conceived under a specific class of priors, such as the point mass mixtures of non-informative and $g$-type priors. The MPM rule, however, has become so very popular that it is now being deployed for a wider variety of priors and under correlated designs, where the properties of MPM are not yet completely understood. The main thrust of this work is to shed light on properties of MPM in these contexts by (a) characterizing situations when MPM is still safe under correlated designs, (b) providing significant generalizations of MPM to a broader class of priors (such as continuous spike-and-slab priors). We also provide new supporting evidence for the suitability of $g$-priors, as opposed to independent product priors, using new predictive matching arguments. Furthermore, we emphasize the importance of prior model probabilities and highlight the merits of non-uniform prior probability assignments using the notion of model aggregates.

This paper addresses the risk of fraud in credit card transactions by developing a probabilistic model for the quickest detection of illegitimate purchases. Using optimal stopping theory, the goal is to determine the moment, known as disorder or fraud time, at which the continuously monitored process of a consumer’s transactions exhibits a disorder due to fraud, in order to return the best trade-off between two sources of cost: on the one hand, the disorder time should be detected as soon as possible to counteract illegal activities and minimize the loss that banks, merchants and consumers suffer; on the other hand, the frequency of false alarms should be minimized to avoid generating adverse effects for cardholders and to limit the operational and process costs for the card issuers. The proposed approach allows us to score consumers’ transactions and to determine, in a rigorous, personalized and optimal manner, the threshold with which scores are compared to establish whether a purchase is fraudulent.

Markov chain Monte Carlo (MCMC) has transformed Bayesian model inference over the past three decades: mainly because of this, Bayesian inference is now a workhorse of applied scientists. Under general conditions, MCMC sampling converges asymptotically to the posterior distribution, but this provides no guarantees about its performance in finite time. The predominant method for monitoring convergence is to run multiple chains and monitor individual chains’ characteristics and compare these to the population as a whole: if within-chain and between-chain summaries are comparable, then this is taken to indicate that the chains have converged to a common stationary distribution. Here, we introduce a new method for diagnosing convergence based on how well a machine learning classifier model can successfully discriminate the individual chains. We call this convergence measure $R^{\ast }$. In contrast to the predominant $\hat{R}$, $R^{\ast }$ is a single statistic across all parameters that indicates lack of mixing, although individual variables’ importance for this metric can also be determined. Additionally, $R^{\ast }$ is not based on any single characteristic of the sampling distribution; instead it uses all the information in the chain, including that given by the joint sampling distribution, which is currently largely overlooked by existing approaches. We recommend calculating $R^{\ast }$ using two different machine learning classifiers — gradient-boosted regression trees and random forests — which each work well in models of different dimensions. Because each of these methods outputs a classification probability, as a byproduct, we obtain uncertainty in $R^{\ast }$. The method is straightforward to implement and could be a complementary additional check on MCMC convergence for applied analyses.

In this paper, we address the numerical posterior error control problem for the Bayesian approach to inverse problems or recently known as Bayesian Uncertainty Quantification (UQ). We generalize the results of () to (a priori) expected Bayes factors (BF) and in a more general, infinite-dimensional setting. In this inverse problem, the unavoidable numerical approximation of the Forward Map (FM, i.e., the regressor function), arising from the numerical solution of a system of differential equations, demands error estimates of the corresponding approximate numerical posterior distribution. Our approach is to make such comparisons in the setting of Bayesian model selection and BFs. The main result of this paper is a bound on the absolute global error tolerated by the numerical solver of the FM in order to keep the BF of the numerical versus the theoretical posterior near one. For two examples, we provide a detailed analysis of the computation and implementation of the introduced bound. Furthermore, we show that the resulting numerical posterior turns out to be nearly identical from the theoretical posterior, given the control of the BF near one.

Women in Colombia face difficulties related to the patriarchal traits of their societies and well-known conflict afflicting the country since 1948. In this critical context, our aim is to study the relationship between baseline socio-demographic factors and variables associated to fertility, partnership patterns, and work activity. To best exploit the explanatory structure, we propose a Bayesian multivariate density regression model, which can accommodate mixed responses with censored, constrained, and binary traits. The flexible nature of the models allows for nonlinear regression functions and non-standard features in the errors, such as asymmetry or multi-modality. The model has interpretable covariate-dependent weights constructed through normalization, allowing for combinations of categorical and continuous covariates. Computational difficulties for inference are overcome through an adaptive truncation algorithm combining adaptive Metropolis-Hastings and sequential Monte Carlo to create a sequence of automatically truncated posterior mixtures. For our study on Colombian women’s life patterns, a variety of quantities are visualised and described, and in particular, our findings highlight the detrimental impact of family violence on women’s choices and behaviors.

We introduce a Bayesian non-parametric spatial factor analysis model with spatial dependency induced through a prior on factor loadings. For each column of the loadings matrix, spatial dependency is encoded using a probit stick-breaking process (PSBP) and a multiplicative gamma process shrinkage prior is used across columns to adaptively determine the number of latent factors. By encoding spatial information into the loadings matrix, meaningful factors are learned that respect the observed neighborhood dependencies, making them useful for assessing rates over space. Furthermore, the spatial PSBP prior can be used for clustering temporal trends, allowing users to identify regions within the spatial domain with similar temporal trajectories, an important task in many applied settings. In the manuscript, we illustrate the model’s performance in simulated data, but also in two real-world examples: longitudinal monitoring of glaucoma and malaria surveillance across the Peruvian Amazon. The R package spBFA, available on CRAN, implements the method.

Bayesian methods have proven themselves to be successful across a wide range of scientific problems and have many well-documented advantages over competing methods. However, these methods run into difficulties for two major and prevalent classes of problems: handling data sets with outliers and dealing with model misspecification. We outline the drawbacks of previous solutions to both of these problems and propose a new method as an alternative. When working with the new method, the data is summarized through a set of insufficient statistics, targeting inferential quantities of interest, and the prior distribution is updated with the summary statistics rather than the complete data. By careful choice of conditioning statistics, we retain the main benefits of Bayesian methods while reducing the sensitivity of the analysis to features of the data not captured by the conditioning statistics. For reducing sensitivity to outliers, classical robust estimators (e.g., M-estimators) are natural choices for conditioning statistics. A major contribution of this work is the development of a data augmented Markov chain Monte Carlo (MCMC) algorithm for the linear model and a large class of summary statistics. We demonstrate the method on simulated and real data sets containing outliers and subject to model misspecification. Success is manifested in better predictive performance for data points of interest as compared to competing methods.

The analysis of rank ordered data has a long history in the statistical literature across a diverse range of applications. In this paper we consider the Extended Plackett-Luce model that induces a flexible (discrete) distribution over permutations. The parameter space of this distribution is a combination of potentially high-dimensional discrete and continuous components and this presents challenges for parameter interpretability and also posterior computation. Particular emphasis is placed on the interpretation of the parameters in terms of observable quantities and we propose a general framework for preserving the mode of the prior predictive distribution. Posterior sampling is achieved using an effective simulation based approach that does not require imposing restrictions on the parameter space. Working in the Bayesian framework permits a natural representation of the posterior predictive distribution and we draw on this distribution to make probabilistic inferences and also to identify potential lack of model fit. The flexibility of the Extended Plackett-Luce model along with the effectiveness of the proposed sampling scheme are demonstrated using several simulation studies and real data examples.

Factor models aim to describe a dependence structure among high-dimensional random variables in terms of a low-dimensional unobserved random vector called a factor. One of the major practical issues of applying the factor model is to determine the factor dimensionality. In this paper, we propose a computationally feasible nonparametric prior distribution which achieves the posterior consistency of the factor dimensionality. We also derive the posterior contraction rate of the covariance matrix which is optimal when the factor dimensionality of the true covariance matrix is bounded. We conduct numerical studies that illustrate our theoretical results.

For the discovery of regression relationships between Y and a large set of p potential predictors $x_1,\dots ,x_p$, the flexible nonparametric nature of BART (Bayesian Additive Regression Trees) allows for a much richer set of possibilities than restrictive parametric approaches. However, subject matter considerations sometimes warrant a minimal assumption of monotonicity in at least some of the predictors. For such contexts, we introduce mBART, a constrained version of BART that can flexibly incorporate monotonicity in any predesignated subset of predictors using a multivariate basis of monotone trees, while avoiding the further confines of a full parametric form. For such monotone relationships, mBART provides (i) function estimates that are smoother and more interpretable, (ii) better out-of-sample predictive performance, and (iii) less post-data uncertainty. While many key aspects of the unconstrained BART model carry over directly to mBART, the introduction of monotonicity constraints necessitates a fundamental rethinking of how the model is implemented. In particular, the original BART Markov Chain Monte Carlo algorithm relied on a conditional conjugacy that is no longer available in a monotonically constrained space. Various simulated and real examples demonstrate the wide ranging potential of mBART.

Global-local shrinkage priors have been recognized as a useful class of priors that can strongly shrink small signals toward prior means while keeping large signals unshrunk. Although such priors have been extensively discussed under Gaussian responses, in practice, we often encounter count responses. Previous contributions on global-local shrinkage priors cannot be readily applied to count data. In this paper, we discuss global-local shrinkage priors for analyzing a sequence of counts. We provide sufficient conditions under which the posterior mean is unshrunk for very large signals, known as the tail robustness property. Then, we propose tractable priors to satisfy those conditions approximately or exactly and develop a custom posterior computation algorithm for Bayesian inference without tuning parameters. We demonstrate the proposed methods through simulation studies and an application to a real dataset.

We study frequentist properties of Bayesian and $L_0$ model selection, with a focus on (potentially non-linear) high-dimensional regression. We propose a construction to study how posterior probabilities and normalized $L_0$ criteria concentrate on the (Kullback-Leibler) optimal model and other subsets of the model space. When such concentration occurs, one also bounds the frequentist probabilities of selecting the correct model, type I and type II errors. These results hold generally, and help validate the use of posterior probabilities and $L_0$ criteria to control frequentist error probabilities associated to model selection and hypothesis tests. Regarding regression, we help understand the effect of the sparsity imposed by the prior or the $L_0$ penalty, and of problem characteristics such as the sample size, signal-to-noise, dimension and true sparsity. A particular finding is that one may use less sparse formulations than would be asymptotically optimal, but still attain consistency and often also significantly better finite-sample performance. We also prove new results related to misspecifying the mean or covariance structures, and give tighter rates for certain non-local priors than currently available.

Many applications in Bayesian statistics are extremely computationally intensive. However, they are often inherently parallel, making them prime targets for modern massively parallel processors. Multi-core and distributed computing is widely applied in the Bayesian community, however, very little attention has been given to fine-grain parallelisation using single instruction multiple data (SIMD) operations that are available on most modern CPUs. In this work, we practically demonstrate, using standard programming libraries, the utility of the SIMD approach for several topical Bayesian applications. Using the C programming language, we show that SIMD can improve the single-core floating point arithmetic performance by up to a factor of $6\times$ compared scalar C code and more than $25\times$ compared with optimised R code. Such improvements are multiplicative to any gains achieved through multi-core processing. We illustrate the potential of SIMD for accelerating Bayesian computations and provide the reader with techniques for exploiting modern massively parallel processing environments.

We consider a binary response which is potentially affected by a set of continuous variables. Of special interest is the causal effect on the response due to an intervention on a specific variable. The latter can be meaningfully determined on the basis of observational data through suitable assumptions on the data generating mechanism. In particular we assume that the joint distribution obeys the conditional independencies (Markov properties) inherent in a Directed Acyclic Graph (DAG), and the DAG is given a causal interpretation through the notion of interventional distribution. We propose a DAG-probit model where the response is generated by discretization through a random threshold of a continuous latent variable and the latter, jointly with the remaining continuous variables, has a distribution belonging to a zero-mean Gaussian model whose covariance matrix is constrained to satisfy the Markov properties of the DAG; the latter is assigned a DAG-Wishart prior through the corresponding Cholesky parameters. Our model leads to a natural definition of causal effect conditionally on a given DAG. Since the DAG which generates the observations is unknown, we present an efficient MCMC algorithm whose target is the posterior distribution on the space of DAGs, the Cholesky parameters of the concentration matrix, and the threshold linking the response to the latent. Our end result is a Bayesian Model Averaging estimate of the causal effect which incorporates parameter, as well as model, uncertainty. The methodology is assessed using simulation experiments and applied to a gene expression data set originating from breast cancer stem cells.

We consider the problem of estimating the predictive density in a heteroskedastic Gaussian model under general divergence loss. Based on a conjugate hierarchical set-up, we consider generic classes of shrinkage predictive densities that are governed by location and scale hyper-parameters. For any α-divergence loss, we propose a risk-estimation based methodology for tuning these shrinkage hyper-parameters. Our proposed predictive density estimators enjoy optimal asymptotic risk properties that are in concordance with the optimal shrinkage calibration point estimation results established by Xie, Kou, and Brown (2012) for heteroskedastic hierarchical models. These α-divergence risk optimality properties of our proposed predictors are not shared by empirical Bayes predictive density estimators that are calibrated by traditional methods such as maximum likelihood and method of moments. We conduct several numerical studies to compare the non-asymptotic performance of our proposed predictive density estimators with other competing methods and obtain encouraging results.

Non-linear hierarchical models are commonly used in many disciplines. However, inference in the presence of non-nested effects and on large datasets is challenging and computationally burdensome. This paper provides two contributions to scalable and accurate inference. First, I derive a new mean-field variational algorithm for estimating binomial logistic hierarchical models with an arbitrary number of non-nested random effects. Second, I propose “marginally augmented variational Bayes” (MAVB) that further improves the initial approximation through a step of Bayesian post-processing. I prove that MAVB provides a guaranteed improvement in the approximation quality at low computational cost and induces dependencies that were assumed away by the initial factorization assumptions.

I apply these techniques to a study of voter behavior using a high-dimensional application of the popular approach of multilevel regression and post-stratification (MRP). Existing estimation took hours whereas the algorithms proposed run in minutes. The posterior means are well-recovered even under strong factorization assumptions. Applying MAVB further improves the approximation by partially correcting the under-estimated variance. The proposed methodology is implemented in an open source software package.

We take a new look at the problem of disentangling the volatility and jumps processes of daily stock returns. We first provide a computational framework for the univariate stochastic volatility model with Poisson-driven jumps that offers a competitive inference alternative to the existing tools. This methodology is then extended to a large set of stocks for which we assume that their unobserved jump intensities co-evolve in time through a dynamic factor model. To evaluate the proposed modelling approach we conduct out-of-sample forecasts and we compare the posterior predictive distributions obtained from the different models. We provide evidence that joint modelling of jumps improves the predictive ability of the stochastic volatility models.

The predictive probabilities of the hierarchical Pitman–Yor process are approximated through Monte Carlo algorithms that exploits the Chinese Restaurant Franchise (CRF) representation. However, in order to simulate the posterior distribution of the hierarchical Pitman–Yor process, a set of auxiliary variables representing the arrangement of customers in tables of the CRF must be sampled through Markov chain Monte Carlo. This paper develops a perfect sampler for these latent variables employing ideas from the Propp–Wilson algorithm and evaluates its average running time by extensive simulations. The simulations reveal a significant dependence of running time on the parameters of the model, which exhibits sharp transitions. The algorithm is compared to simpler Gibbs sampling procedures, as well as a procedure for unbiased Monte Carlo estimation proposed by Glynn and Rhee. We illustrate its use with an example in microbial genomics studies.

Actin cytoskeleton networks generate local topological signatures due to the natural variations in the number, size, and shape of holes of the networks. Persistent homology is a method that explores these topological properties of data and summarizes them as persistence diagrams. In this work, we analyze and classify simulated actin filament networks by transforming them into persistence diagrams whose variability is quantified via a Bayesian framework on the space of persistence diagrams. The proposed generalized Bayesian framework adopts an independent and identically distributed cluster point process characterization of persistence diagrams and relies on a substitution likelihood argument. This framework provides the flexibility to estimate the posterior cardinality distribution of points in a persistence diagram and their posterior spatial distribution simultaneously. We present a closed form of the posteriors under the assumption of a Gaussian mixture and binomial for prior intensity and cardinality respectively. Using this posterior calculation, finally, we implement a Bayes factor algorithm to classify simulated actin filament networks and benchmark it against several state-of-the-art classification methods.

Conditional heteroscedastic (CH) models are routinely used to analyze financial datasets. The classical models such as ARCH-GARCH with time-invariant coefficients are often inadequate to describe frequent changes over time due to market variability. However, we can achieve significantly better insight by considering the time-varying analogs of these models. In this paper, we propose a Bayesian approach to the estimation of such models and develop a computationally efficient MCMC algorithm based on Hamiltonian Monte Carlo (HMC) sampling. We also established posterior contraction rates with increasing sample size in terms of the average Hellinger metric. The performance of our method is compared with frequentist estimates and estimates from the time constant analogs. To conclude the paper we obtain time-varying parameter estimates for some popular Forex (currency conversion rate) and stock market datasets.