Estimating heterogeneous treatment effects has drawn increasing attention in medical studies, considering that patients with divergent features can undergo a different progression of disease even with identical treatment. Such heterogeneity can co-occur with a cured fraction for biomedical studies with a time-to-event outcome and further complicates the quantification of treatment effects. This study considers a joint framework of Bayesian causal forest and accelerated failure time cure model to capture the cured proportion and treatment effect heterogeneity through three separate Bayesian additive regression trees. Under the potential outcomes framework, conditional and sample average treatment effects within the uncured subgroup are derived on the scale of log survival time subject to right-censoring, and treatment effects on the scale of survival probability are derived for each individual. Bayesian backfitting Markov chain Monte Carlo algorithm with the Gibbs sampler is conducted to estimate the causal effects. Simulation studies show the satisfactory performance of the proposed method. The proposed model is then applied to a breast cancer dataset extracted from the SEER database to demonstrate its usage in detecting heterogeneous treatment effects and cured subgroups. Combined with popular mitigation strategies, the proposed method can also alleviate confounding induced by immortal time bias.

Differential equations based on physical principals are used to represent complex dynamic systems in all fields of science and engineering. When known, these equations have been shown to well represent real-world dynamics. However, since the true dynamics of complex systems are generally unknown, learning the governing equations can improve our understanding of the mechanisms driving the systems. Here, we develop a Bayesian approach to data-driven discovery of nonlinear spatio-temporal dynamic equations. Our approach can accommodate measurement error and missing data, both of which are common in real-world data, and accounts for parameter uncertainty. The proposed framework is illustrated using three simulated systems with varying amounts of measurement uncertainty and missing data and applied to a real-world system to infer the temporal evolution of the vorticity of the streamfunction.

Interactions between actors are frequently represented using a network. The latent position model is widely used for analysing network data, whereby each actor is positioned in a latent space. Inferring the dimension of this space is challenging. Often, for simplicity, two dimensions are used or model selection criteria are employed to select the dimension, but this requires choosing a criterion and the computational expense of fitting multiple models. Here the latent shrinkage position model (LSPM) is proposed which intrinsically infers the effective dimension of the latent space. The LSPM employs a Bayesian nonparametric multiplicative truncated gamma process prior that ensures shrinkage of the variance of the latent positions across higher dimensions. Dimensions with non-negligible variance are deemed most useful to describe the observed network, inducing automatic inference on the latent space dimension. While the LSPM is applicable to many network types, logistic and Poisson LSPMs are developed here for binary and count networks respectively. Inference proceeds via a Markov chain Monte Carlo algorithm, where novel surrogate proposal distributions reduce the computational burden. The LSPM’s properties are assessed through simulation studies, and its utility is illustrated through application to real network datasets. Open source software assists wider implementation of the LSPM.

Markov chain Monte Carlo methods for exponential family models with intractable normalizing constant, such as the exchange algorithm, require simulations of the sufficient statistics at every iteration of the Markov chain, which often result in expensive computations. Surrogate models for the likelihood function have been developed to accelerate inference algorithms in this context. However, these surrogate models tend to be relatively inflexible, and often provide a poor approximation to the true likelihood function. In this article, we propose the use of a warped, gradient-enhanced, Gaussian process surrogate model for the likelihood function, which jointly models the sample means and variances of the sufficient statistics, and uses warping functions to capture covariance nonstationarity in the input parameter space. We show that both the consideration of nonstationarity and the inclusion of gradient information can be leveraged to obtain a surrogate model that outperforms the conventional stationary Gaussian process surrogate model when making inference, particularly in regions where the likelihood function exhibits a phase transition. We also show that the proposed surrogate model can be used to improve the effective sample size per unit time when embedded in exact inferential algorithms. The utility of our approach in speeding up inferential algorithms is demonstrated on simulated and real-world data.

Simulation-based calibration checking (SBC) is a practical method to validate computationally-derived posterior distributions or their approximations. In this paper, we introduce a new variant of SBC to alleviate several known problems. Our variant allows the user to in principle detect any possible issue with the posterior, while previously reported implementations could never detect large classes of problems including when the posterior is equal to the prior. This is made possible by including additional data-dependent test quantities when running SBC. We argue and demonstrate that the joint likelihood of the data is an especially useful test quantity. Some other types of test quantities and their theoretical and practical benefits are also investigated. We provide theoretical analysis of SBC, thereby providing a more complete understanding of the underlying statistical mechanisms. We also bring attention to a relatively common mistake in the literature and clarify the difference between SBC and checks based on the data-averaged posterior. We support our recommendations with numerical case studies on a multivariate normal example and a case study in implementing an ordered simplex data type for use with Hamiltonian Monte Carlo. The SBC variant introduced in this paper is implemented in the SBC R package.

DNA methylation datasets in cancer studies are comprised of measurements on a large number of genomic locations called cytosine-phosphate-guanine (CpG) sites with complex correlation structures. A fundamental goal of these studies is the development of statistical techniques that can identify disease genomic signatures across multiple patient groups defined by different experimental or biological conditions. We propose BayesDiff, a nonparametric Bayesian approach for differential analysis relying on a novel class of first order mixture models called the Sticky Pitman-Yor process or two-restaurant two-cuisine franchise (2R2CF). The BayesDiff methodology flexibly utilizes information from all CpG sites or biomarker probes, adaptively accommodates any serial dependence due to the widely varying inter-probe distances, and makes posterior inferences about the differential genomic signature of patient groups. Using simulation studies, we demonstrate the effectiveness of the BayesDiff procedure relative to existing statistical techniques for differential DNA methylation. The methodology is applied to analyze a gastrointestinal (GI) cancer dataset exhibiting serial correlation and complex interaction patterns. The results support and complement known aspects of DNA methylation and gene association in upper GI cancers.

We consider the problem of performing inference for a continuous treatment effect on a binary outcome variable while controlling for high dimensional baseline covariates. We propose a novel Bayesian framework for performing inference for the desired low-dimensional parameter in a high-dimensional logistic regression model. While it is relatively easier to address this problem in linear regression, the nonlinearity of the logistic regression poses additional challenges that make it difficult to orthogonalize the effect of the treatment variable from the nuisance variables. Our proposed approach provides the first Bayesian alternative to the recent frequentist developments and can incorporate available prior information on the parameters of interest, which plays a crucial role in practical applications. In addition, the proposed approach incorporates uncertainty in orthogonalization in high dimensions instead of relying on a single instance of orthogonalization as done by frequentist methods. We provide uniform convergence results that show the validity of credible intervals resulting from the posterior. Our method has competitive empirical performance when compared with state-of-the-art methods.

Normalized random measures with independent increments (NRMIs) represent a large class of Bayesian nonparametric priors and are widely used in the Bayesian nonparametric framework. In this paper, we provide the posterior consistency analysis for these NRMIs through their characterizing Lévy intensities. Assumptions are introduced on the Lévy intensities to analyse the posterior consistency and are verified with multiple interesting examples. Another focus of the paper is the Bernstein-von Mises theorem for a particular subclass of NRMIs, namely the normalized generalized gamma processes (NGGP). When the Bernstein-von Mises theorem is applied to construct credible sets, in addition to the usual form, there will be an additional bias term on the left endpoint closely related to the number of atoms of the true distribution in the discrete case. We also discuss the effect of the estimators for the model parameters of the NGGP under the Bernstein-von Mises convergence. Finally, to further illustrate the impact of the bias correction term in the construction of credible sets, we present a numerical example to demonstrate numerically how the bias correction affects the coverage of the true value.

Bayesian cross-validation (CV) is a popular method for predictive model assessment that is simple to implement and broadly applicable. A wide range of CV schemes is available for time series applications, including generic leave-one-out (LOO) and K-fold methods, as well as specialized approaches intended to deal with serial dependence such as leave-future-out (LFO), h-block, and $hv$-block.

Existing large-sample results show that both specialized and generic methods are applicable to models of serially-dependent data. However, large sample consistency results overlook the impact of sampling variability on accuracy in finite samples. Moreover, the accuracy of a CV scheme depends on many aspects of the procedure. We show that poor design choices can lead to elevated rates of adverse selection.

In this paper, we consider the problem of identifying the regression component of an important class of models of data with serial dependence, autoregressions of order p with q exogenous regressors ($\mathsf{ARX}(p,q)$), under the logarithmic scoring rule. We show that when serial dependence is present, scores computed using the joint (multivariate) density have lower variance and better model selection accuracy than the popular pointwise estimator. In addition, we present a detailed case study of the special case of $\mathsf{ARX}$ models with fixed autoregressive structure and variance. For this class, we derive the finite-sample distribution of the CV estimators and the model selection statistic. We conclude with recommendations for practitioners.

Mobile health (mHealth) methods allow researchers to monitor study participants in their natural environments in order to improve health-related outcomes through behavior change. MHealth investigators are interested in understanding the feasibility of supplementing or even replacing actively collected data with passively collected data to reduce participant burden, while also increasing the temporal resolution of the data. In this work, we propose a novel Bayesian dynamic functional variable selection method to explore the relations between multimodal mHealth data collected on different time scales. Specifically, our approach leverages spiked hierarchical species sampling priors to identify critical moments when a participant experiences momentary spikes in a low-frequency outcome, characterize high-frequency outcome trajectories which are related to these critical moments, and cluster the relational trends to explore potential subpopulations. We introduce continuous-time multistate Markov model priors to inform selection based on information learned at previous assessments. We demonstrate the variable selection and clustering performance of our model in various simulation settings motivated by the data structures found in mHealth studies. We then apply our model to multimodal intensive longitudinal data collected in the Pathways between Socioeconomic Status and Behavioral Cancer Risk Factors Study to explore relations between physical activity passively collected with accelerometers and mood actively collected with ecological momentary assessment methods.

In environmental health research there is often interest in the effect of an exposure on a health outcome assessed on the same day and several subsequent days or lags. Distributed lag nonlinear models (DLNM) are a well-established statistical framework for estimating an exposure-lag-response function. We propose methods to allow for prior information to be incorporated into DLNMs. First, we impose a monotonicity constraint in the exposure-response at lagged time periods which matches with knowledge on how biological mechanisms respond to increased levels of exposures. Second, we introduce variable selection into the DLNM to identify lagged periods of susceptibility with respect to the outcome of interest. The variable selection approach allows for direct application of informative priors on which lags have nonzero association with the outcome. We propose a tree-of-trees model that uses two layers of trees: one for splitting the exposure time frame and one for fitting exposure-response functions over different time periods. We introduce a zero-inflated alternative to the tree splitting prior in Bayesian additive regression trees to allow for lag selection and the addition of informative priors. We develop a computational approach for efficient posterior sampling and perform a comprehensive simulation study to compare our method to existing DLNM approaches. We apply our method to estimate time-lagged extreme temperature relationships with mortality during summer or winter in Chicago, IL.

It is common to hold prior beliefs that are not characterized by points in the parameter space but instead are relational in nature and can be described by a linear subspace. While some previous work has been done to account for such prior beliefs, the focus has primarily been on point estimators within a regression framework. We argue, however, that prior beliefs about parameters ought to be encoded into the prior distribution rather than in the formation of a point estimator. In this way, the prior beliefs help shape all inference. Through exponential tilting, we propose a fully generalizable method of taking existing prior information from, e.g., a pilot study, and combining it with additional prior beliefs represented by parameters lying on a linear subspace. We provide computationally efficient algorithms for posterior inference that, once inference is made using a non-tilted prior, does not depend on the sample size. We illustrate our proposed approach on an antihypertensive clinical trial dataset where we shrink towards a power law dose-response relationship, and on monthly influenza and pneumonia data where we shrink moving average lag parameters towards smoothness. Software to implement the proposed approach is provided in the R package SUBSET available on GitHub.

Bayesian nonparametric mixture models offer a rich framework for model-based clustering. We consider the situation where the kernel of the mixture is available only up to an intractable normalizing constant. In this case, the most commonly used Markov chain Monte Carlo (MCMC) methods are unsuitable. We propose an approximate Bayesian computational (ABC) strategy, whereby we approximate the posterior to avoid the intractability of the kernel. We derive an ABC-MCMC algorithm which combines (i) the use of the predictive distribution induced by the nonparametric prior as proposal and (ii) the use of the Wasserstein distance and its connection to optimal matching problems. To overcome the sensibility concerning the parameters of our algorithm, we further propose an adaptive strategy. We illustrate the use of the proposed algorithm with several simulation studies and an application on real data, where we cluster a population of networks, comparing its performance with standard MCMC algorithms and validating the adaptive strategy.

Species-sampling problems (SSPs) refer to a vast class of statistical problems calling for the estimation of (discrete) functionals of the unknown species composition of an unobservable population. A common feature of SSPs is their invariance with respect to species labeling, which is at the core of the Bayesian nonparametric (BNP) approach to SSPs under the popular Pitman-Yor process (PYP) prior. In this paper, we develop a BNP approach to SSPs that are not “invariant” to species labeling, in the sense that an ordering or ranking is assigned to species’ labels. Inspired by the population genetics literature on age-ordered alleles’ compositions, we study the following SSP with ordering: given an observable sample from an unknown population of individuals belonging to species (alleles), with species’ labels being ordered according to weights (ages), estimate the frequencies of the first r order species’ labels in an enlarged sample obtained by including additional unobservable samples. By relying on an ordered PYP prior, we obtain an explicit posterior distribution of the first r order frequencies, with estimates being of easy implementation and computationally efficient. We apply our approach to the analysis of genetic variation, showing its effectiveness in estimating the frequency of the oldest allele, and then we discuss other potential applications.

The increasing prevalence of network data in a vast variety of fields and the need to extract useful information out of them have spurred fast developments in related models and algorithms. Among the various learning tasks with network data, community detection, the discovery of node clusters or “communities,” has arguably received the most attention in the scientific community. In many real-world applications, the network data often come with additional information in the form of node or edge covariates that should ideally be leveraged for inference. In this paper, we add to a limited literature on community detection for networks with covariates by proposing a Bayesian stochastic block model with a covariate-dependent random partition prior. Under our prior, the covariates are explicitly expressed in specifying the prior distribution on the cluster membership. Our model has the flexibility of modeling uncertainties of all the parameter estimates including the community membership. Importantly, and unlike the majority of existing methods, our model has the ability to learn the number of the communities via posterior inference without having to assume it to be known. Our model can be applied to community detection in both dense and sparse networks, with both categorical and continuous covariates, and our MCMC algorithm is very efficient with good mixing properties. We demonstrate the superior performance of our model over existing models in a comprehensive simulation study and an application to two real datasets.

Dirichlet processes and their extensions have reached a great popularity in Bayesian nonparametric statistics. They have also been introduced for spatial and spatio-temporal data, as a tool to analyze and predict surfaces. A popular approach to Dirichlet processes in a spatial setting relies on a stick-breaking representation of the process, where the dependence over space is described in the definition of the stick-breaking probabilities. Extensions to include temporal dependence are still limited, however it is important, in particular for those phenomena which may change rapidly over time and space, with many local changes. In this work, we propose a Dirichlet process where the stick-breaking probabilities are defined to incorporate both spatial and temporal dependence. We will show that this approach is not a simple extension of available methodologies and can outperform available approaches in terms of prediction accuracy. The approach results in a predictive model that does not rely on the assumptions of Gaussianity or the separability of time and space.

In this paper, we study the finite approximation of the completely random measure (CRM) by truncating its Ferguson-Klass representation. The approximation is obtained by keeping the N largest atom weights of the CRM unchanged and combining the smaller atom weights into a single term. We develop the simulation algorithms for the approximation and characterise its posterior distribution, for which a blocked Gibbs sampler is devised. We demonstrate the usage of the approximation in two models. The first assumes such an approximation as the mixing distribution of a Bayesian nonparametric mixture model and leads to a finite approximation to the model posterior. The second concerns the finite approximation to the Caron-Fox model. Examples and numerical implementations are given based on the gamma, stable and generalised gamma processes.

Heavy tails are often found in practice, and yet they are an Achilles heel of a variety of mainstream random probability measures such as the Dirichlet process (DP). The first contribution of this paper focuses on characterizing the tails of the so-called normalized generalized gamma (NGG) process. We show that the right tail of an NGG process is heavy-tailed provided that the centering distribution is itself heavy-tailed; the DP is the only member of the NGG class that fails to obey this convenient property. A second contribution of the paper rests on the development of two classes of heavy-tailed mixture models and the assessment of their relative merits. Multivariate extensions of the proposed heavy-tailed mixtures are devised here, along with a predictor-dependent version, to learn about the effect of covariates on a multivariate heavy-tailed response. The simulation study suggests that the proposed method performs well in various scenarios, and we showcase the application of the proposed methods in a neuroscience dataset.

In recent years, new cancer treatments have improved survival in multiple histologies. Some of these therapeutics, and in particular treatment combinations, are often associated with severe treatment-related adverse events (AEs). Therefore, It is important to identify alternative de-intensified therapies, such as dose-reduced therapies with reduced AEs and similar efficacy. We introduce a sequential design for multi-arm de-intensification studies. Based on joint modeling of toxicity and efficacy endpoints, the design evaluates multiple de-intensified therapies at different dose levels, one at a time. We study the utility of the design in oropharynx cancer de-intensification studies. We use a joint Bayesian model for efficacy and toxicity outcomes to define decision rules at interim and final analyses. Interim decisions include early termination of the study due to inferior survival of experimental arms compared to a standard of care (SOC), and the transitions from one de-intensified treatment arm to another with a further reduced dose when there is sufficient evidence of non-inferior survival. We evaluate the operating characteristics of the design using data from recent de-intensification studies in oropharynx cancer.

There has been increased research interest in the subfield of sparse Bayesian factor analysis with shrinkage priors, which achieve additional sparsity beyond the natural parsimonity of factor models. In this spirit, we estimate the number of common factors in the widely applied sparse latent factor model with spike-and-slab priors on the factor loadings matrix. Our framework leads to a natural, efficient and simultaneous coupling of model estimation and selection on one hand and model identification and rank estimation (number of factors) on the other hand. More precisely, by embedding the unordered generalised lower triangular loadings representation into overfitting sparse factor modelling, we obtain posterior summaries regarding factor loadings, common factors as well as the factor dimension via postprocessing draws from our efficient and customized Markov chain Monte Carlo scheme.

Clinical trials often allow patients in the control arm to switch to the treatment arm if their physical conditions are worse than certain tolerance levels. For instance, treatment switching arises in the Concorde clinical trial, which aims to assess causal effects on the time-to-disease progression or death of immediate versus deferred treatment with zidovudine among patients with asymptomatic HIV infection. The Intention-To-Treat analysis does not measure the effect of the actual receipt of the treatment and ignores the information on treatment switching. Other existing methods reconstruct the outcome a patient would have had if they had not switched under strong assumptions. Departing from the literature, we re-define the problem of treatment switching using principal stratification and focus on causal effects for patients belonging to subpopulations defined by the switching behavior under control. We use a Bayesian approach to inference, taking into account that $(i)$ switching happens in continuous time; $(ii)$ switching time is not defined for patients who never switch in a particular experiment; and $(iii)$ survival time and switching time are subject to censoring. We apply this framework to analyze the synthetic data based on the Concorde study.

Additive spatial statistical models with weakly stationary process assumptions have become standard in spatial statistics. However, one disadvantage of such models is the computation time, which rapidly increases with the number of data points. The goal of this article is to apply an existing subsampling strategy to standard spatial additive models and to derive the spatial statistical properties. We call this strategy the “spatial data subset model” (SDSM) approach, which can be applied to big datasets in a computationally feasible way. Our approach has the advantage that one does not require any additional restrictive model assumptions. That is, computational gains increase as model assumptions are removed when using our model framework. This provides one solution to the computational bottlenecks that occur when applying methods such as Kriging to “big data”. We provide several properties of this new spatial data subset model approach in terms of moments, sill, nugget, and range under several sampling designs. An advantage of our approach is that it subsamples without throwing away data, and can be implemented using datasets of any size that can be stored. We present the results of the spatial data subset model approach on simulated datasets, and on a large dataset consists of 150,000 observations of daytime land surface temperatures measured by the MODIS instrument onboard the Terra satellite.

As a regression technique in spatial statistics, the spatiotemporally varying coefficient model (STVC) is an important tool for discovering nonstationary and interpretable response-covariate associations over both space and time. However, it is difficult to apply STVC for large-scale spatiotemporal analyses due to its high computational cost. To address this challenge, we summarize the spatiotemporally varying coefficients using a third-order tensor structure and propose to reformulate the spatiotemporally varying coefficient model as a special low-rank tensor regression problem. The low-rank decomposition can effectively model the global patterns of large data sets with a substantially reduced number of parameters. To further incorporate the local spatiotemporal dependencies, we use Gaussian process (GP) priors on the spatial and temporal factor matrices. We refer to the overall framework as Bayesian Kernelized Tensor Regression (BKTR), and kernelized tensor factorization can be considered a new and scalable approach to modeling multivariate spatiotemporal processes with a low-rank covariance structure. For model inference, we develop an efficient Markov chain Monte Carlo (MCMC) algorithm, which uses Gibbs sampling to update factor matrices and slice sampling to update kernel hyperparameters. We conduct extensive experiments on both synthetic and real-world data sets, and our results confirm the superior performance and efficiency of BKTR for model estimation and parameter inference.

Within the field of record linkage, Bayesian methods have the crucial advantage of quantifying uncertainty from imperfect linkages. However, current implementations of Bayesian Fellegi-Sunter models are computationally intensive, making them challenging to use on larger-scale record linkage tasks. To address these computational difficulties, we propose fast beta linkage (fabl), an extension to the Beta Record Linkage (BRL) method of Sadinle (2017). Specifically, we use independent prior distributions over the matching space, allowing us to use hashing techniques that reduce computational overhead. This also allows us to complete pairwise record comparisons over large data files through parallel computing and to reduce memory costs through a new technique called storage efficient indexing. Through simulations and two case studies, we show that fabl can have markedly increased speed with minimal loss of accuracy when compared to BRL.

Due to their flexibility Gaussian processes are a well-known Bayesian framework for nonparametric function estimation. Integrating inequality constraints, such as monotonicity, convexity, and boundedness, into Gaussian process models significantly improves prediction accuracy and yields more realistic credible intervals in various real-world data applications. The finite-dimensional Gaussian process approximation, originally proposed in Maatouk and Bay (2017) is considered. This method involves approximating a parent GP by utilizing a finite-dimensional GP obtained through appropriate basis expansions. It satisfies interpolation conditions and handles a wide range of inequality constraints everywhere. Our contribution in this paper is threefold. First, we extend this approach to handle noisy observations and multiple, more general convex and non-convex constraints. Second, we propose new basis functions in order to extend the smoothness of sample paths to differentiability of class $C^p$, for any $p\ge 1$. Third, we examine its behavior in specific scenarios such as monotonicity with flat regions and boundedness near lower and/or upper bounds. In that case, we show that, unlike the Maximum a posteriori (MAP) estimate, the mean a posteriori (mAP) estimate fails to capture flat regions. To address this issue, we propose incorporating multiple constraints, such as monotonicity with bounded slope constraints. According to the theoretical convergence and based on a variety of numerical experiments, the MAP estimate behaves well and outperforms the mAP estimate in terms of prediction accuracy. The performance of the proposed approach is confirmed through synthetic and real-world data studies.

We propose an easily computed estimator of the marginal likelihood from posterior simulation output, via reciprocal importance sampling, combining earlier proposals of DiCiccio et al (1997) and Robert and Wraith (2009). This involves only the unnormalized posterior densities from the sampled parameter values, and does not involve additional simulations beyond the main posterior simulation, or additional complicated calculations, provided that the parameter space is unconstrained. Even if this is not the case, the estimator is easily adjusted by a simple Monte Carlo approximation. It is unbiased for the reciprocal of the marginal likelihood, consistent, has finite variance, and is asymptotically normal. It involves one user-specified control parameter, and we derive an optimal way of specifying this. We illustrate it with several numerical examples.

The multivariate regression interpretation of the Gaussian chain graph model simultaneously parametrizes (i) the direct effects of p predictors on q outcomes and (ii) the residual partial covariances between pairs of outcomes. We introduce a new method for fitting sparse versions of these models with spike-and-slab LASSO (SSL) priors. We develop an Expectation Conditional Maximization algorithm to obtain sparse estimates of the $p\times q$ matrix of direct effects and the $q\times q$ residual precision matrix. Our algorithm iteratively solves a sequence of penalized maximum likelihood problems with self-adaptive penalties that gradually filter out negligible regression coefficients and partial covariances. Because it adaptively penalizes individual model parameters, our method is seen to outperform fixed-penalty competitors on simulated data. We establish the posterior contraction rate for our model, buttressing our method’s excellent empirical performance with strong theoretical guarantees. Using our method, we estimated the direct effects of diet and residence type on the composition of the gut microbiome of elderly adults.

A conventional approach to the extraction of latent components in a time series is to first model extreme values (including level shifts and seasonal outliers) as fixed effects, followed by their removal. Then the extreme-value adjusted series can be filtered using linear (Gaussian) techniques. A drawback is that identification of the epochs of extreme values is needed, and the uncertainty about this identification – as well as the removal of extremes – goes unmeasured. Alternatively, each outlier effect can be modeled as a particular type of latent stochastic process driven by heavy-tailed innovations; extraction of latent components then follows non-linear techniques and does not require identification of extreme epochs. We model monthly retail data impacted by the Covid-19 epidemic by incorporating additive outliers and level shifts as heavy-tailed latent processes, and estimate the unknown parameters through a Bayesian approach that utilizes Gibbs sampling. As a result, we can extract retail trends that incorporate stochastic level shifts and a full measure of the extraction uncertainty. An added benefit of the proposed approach is an estimate of a counterfactual trend following an extreme event. The posterior estimate of the counterfactual trend can be used to quantify the impact of an extreme event.

The Matérn family of covariance functions plays a prominent role in the analysis of geostatistical data due to its ability to model different smoothness behaviors. Although in many applications the smoothness parameter is set at an arbitrary value, a more satisfactory approach requires data–based inference about smoothness, especially in the light of new findings showing that the information this type of data has about the smoothness can be considerable in some settings. This work proposes a new class of easy–to–compute default priors for the parameters of a class of Gaussian random fields with Matérn covariance functions with unknown smoothness parameter. This class of priors is obtained by approximating a reference prior using the spectral representation of stationary random fields. This approximate reference prior has several advantages over the exact reference prior. First, the computation of the former is more stable and considerably less burdensome than that of the latter. Second, both the marginal prior of the smoothness parameter and the joint posterior of all parameters are proper for the approximate reference prior, while the status of these for the exact reference prior is currently unknown. Third, Bayesian inferences about the covariance parameters based on the approximate reference prior have satisfactory frequentist properties that are superior than those based on maximum likelihood. It was also found that a previously proposed ad–hoc prior for the smoothness and the approximate reference prior display in most cases similar statistical performance, with the former being computationally simpler. The methodology is illustrated with an analysis of rainfall totals in Switzerland.

Bayesian inference for a simple generative model for rank ordered data with ties is considered. The model is based on ordering geometric latent variables and can be seen as the discrete counterpart of the Plackett-Luce (PL) model, a popular, relatively tractable model for permutations. The model, which will be referred to as the GPL model, for generalized (or geometric) Plackett-Luce model, contains the PL model as a limiting special case. A closed form expression for the likelihood is derived. With a focus on Bayesian inference via data augmentation, simple Gibbs sampling and EM algorithms are derived for both the general case of multiple comparisons and the special case of paired comparisons. The methodology is applied to several real data examples. The examples highlight the flexibility of the GPL model to cope with a range of data types, the simplicity and efficiency of the inferential algorithms, and the ability of the GPL model to naturally facilitate predictive inference due to its simple generative construction.

Latent Class Models (LCMs) are used to cluster multivariate categorical data (e.g. group participants based on survey responses). Traditional LCMs assume a property called conditional independence. This assumption can be restrictive, leading to model misspecification and overparameterization. To combat this problem, we developed a novel Bayesian model called a Domain Latent Class Model (DLCM), which permits conditional dependence. We verify identifiability of DLCMs. We also demonstrate the effectiveness of DLCMs in both simulations and real-world applications. Compared to traditional LCMs, DLCMs are effective in applications with time series, overlapping items, and structural zeroes.

Chain event graphs have been established as a practical Bayesian graphical tool. While bespoke diagnostics have been developed for Bayesian Networks, they have not yet been defined for the statistical class of Chain Event Graph models. Mirroring the methodology of prequential diagnostics for Bayesian Networks, in this paper we design a number of new Bayesian diagnostics for this new class. These can be used to check whether a selected model – presumably the best within the class – captures most of the salient features of the observed process. These are designed to check the continued validity of a selected model as data about a population is collected. A previous study of health care outcomes illustrates the efficacy of these diagnostics. A second example on breast cancer recurrence offers a more expressive example.

Multi-group covariance estimation for matrix-variate data with small within-group sample sizes is a key part of many data analysis tasks in modern applications. To obtain accurate group-specific covariance estimates, shrinkage estimation methods which shrink an unstructured, group-specific covariance either across groups towards a pooled covariance or within each group towards a Kronecker structure have been developed. However, in many applications, it is unclear which approach will result in more accurate covariance estimates. In this article, we present a hierarchical prior distribution which flexibly allows for both types of shrinkage. The prior linearly combines shrinkage across groups towards a shared pooled covariance and shrinkage within groups towards a group-specific Kronecker covariance. We illustrate the utility of the proposed prior in speech recognition and an analysis of chemical exposure data.

Point estimation is a fundamental statistical task. Given the wide selection of available point estimators, it is unclear, however, what, if any, would be universally-agreed theoretical reasons to generally prefer one such estimator over another. In this paper, we define a class of estimation scenarios which includes commonly-encountered problem situations such as both “high stakes” estimation and scientific inference, and introduce a new class of estimators, Error Intolerance Candidates (EIC) estimators, which we prove is optimal for it.

EIC estimators are parameterised by an externally-given loss function. We prove, however, that even without such a loss function if one accepts a small number of incontrovertible-seeming assumptions regarding what constitutes a reasonable loss function, the optimal EIC estimator can be characterised uniquely.

The optimal estimator derived in this second case is a previously-studied combination of maximum a posteriori (MAP) estimation and Wallace-Freeman (WF) estimation which has long been advocated among Minimum Message Length (MML) researchers, where it is derived as an approximation to the information-theoretic Strict MML estimator. Our results provide a novel justification for it that is purely Bayesian and requires neither approximations nor coding, placing both MAP and WF as special cases in the larger class of EIC estimators.

Sequential algorithms such as sequential importance sampling (SIS) and sequential Monte Carlo (SMC) have proven fundamental in Bayesian inference for models not admitting a readily available likelihood function. For approximate Bayesian computation (ABC), SMC-ABC is the state-of-art sampler. However, since the ABC paradigm is intrinsically wasteful, sequential ABC schemes can benefit from well-targeted proposal samplers that efficiently avoid improbable parameter regions. We contribute to the ABC modeller’s toolbox with novel proposal samplers that are conditional to summary statistics of the data. In a sense, the proposed parameters are “guided” to rapidly reach regions of the posterior surface that are compatible with the observed data. This speeds up the convergence of these sequential samplers, thus reducing the computational effort, while preserving the accuracy in the inference. We provide a variety of guided Gaussian and copula-based samplers for both SIS-ABC and SMC-ABC easing inference for challenging case-studies, including multimodal posteriors, highly correlated posteriors, hierarchical models with about 20 parameters, and a simulation study of cell movements using more than 400 summary statistics.

We propose a Bayesian joint bandwidth selection method for estimating banded Cholesky factors from multiple groups, where variables have a natural ordering and only depend on their predecessors or neighborhoods. The proposed prior is tailored to the local dependence structure and assumes that different groups share a similar structure. We show that the resulting posterior achieves joint selection consistency and nearly optimal posterior convergence rates in high-dimensional settings. Furthermore, the proposed method enables scalable inference without Markov chain Monte Carlo by providing a closed posterior, which guarantees fast computational speed even for large datasets. The practical performance of the method is demonstrated through synthetic and real data analysis.

Bayesian inference in latent Gaussian models necessitates the specification of prior distributions for scale parameters, which govern the behavior of model components. This task is particularly delicate and many contributions in the literature are devoted to the topic. We show that the scale parameter plays a crucial role in determining the prior variability of the model components, which is influenced by factors such as correlation structure, design matrices, and potential linear constraints. This intricate relationship adds complexity, making it difficult to interpret and compare priors across diverse applications. To tackle this challenge, we propose a novel approach for prior specification based on the theory of distribution of quadratic forms. Our strategy involves the use of design and structure-dependent (DSD) priors, which ensure a consistent interpretation across diverse applications. By introducing a single parameter that governs the prior variability of the linear predictor, we simplify the process of prior specification, making it more manageable and interpretable. We derive analytical expressions for DSD priors on scale parameters and establish conditions that guarantee their existence. To demonstrate the efficacy of our proposed prior elicitation strategy, we conduct a simulation study, examining the sampling properties of the estimators. Additionally, we explore several real data applications to investigate prior sensitivity and the allocation of explained variance among model components.

Recent developments in parallel Markov chain Monte Carlo (MCMC) algorithms allow us to run thousands of chains almost as quickly as a single chain, using hardware accelerators such as GPUs. While each chain still needs to forget its initial point during a warmup phase, the subsequent sampling phase can be shorter than in classical settings, where we run only a few chains. To determine if the resulting short chains are reliable, we need to assess how close the Markov chains are to their stationary distribution after warmup. The potential scale reduction factor $\hat{R}$ is a popular convergence diagnostic but unfortunately can require a long sampling phase to work well. We present a nested design to overcome this challenge and a generalization called nested $\hat{R}$. This new diagnostic works under conditions similar to $\hat{R}$ and completes the workflow for GPU-friendly samplers. In addition, the proposed nesting provides theoretical insights into the utility of $\hat{R}$, in both classical and short-chains regimes.

We study scoring rules to assess forecasts of trajectories, given either in discrete or continuous time. Our approach leverages the statistical framework of proper scoring rules with classical mathematical results to derive a principled approach to decision making with such forecasts. In particular, we introduce notions of gradients, entropy, and divergence that are tailor-made to respect the underlying non-Euclidean structure. This results in computable quantities and we demonstrate their efficiency in classical inference tasks such as two-sample and independence testing.

This work considers Bayesian inference under misspecification for complex statistical models comprised of simpler submodels, referred to as modules, that are coupled together. Such “multi-modular” models often arise when combining information from different data sources where there is a module for each data source. When some of the modules are misspecified, the challenges of Bayesian inference under misspecification can sometimes be addressed by using “cutting feedback” methods, which modify conventional Bayesian inference by limiting the influence of unreliable modules. Here we investigate cutting feedback methods in the context of generalized posterior distributions built from loss functions. We make three main contributions. First, we describe how cutting feedback methods can be defined in the generalized Bayes setting, and discuss the appropriate scaling of the loss functions in this context. Second, we derive a novel type of conditional Laplace approximation that accurately describes the behavior of the posterior for a given module’s parameters when we condition on parameters in other modules. Third, we leverage this novel result to provide several convenient diagnostics for Bayesian modular inference, which we then apply to examples in the literature on cut model inference.

We consider estimation of a normal mean matrix under the Frobenius loss. Motivated by the Efron–Morris estimator, a generalization of Stein’s prior has been recently developed, which is superharmonic and shrinks the singular values towards zero. The generalized Bayes estimator with respect to this prior is minimax and dominates the maximum likelihood estimator. However, here we show that it is inadmissible by using Brown’s condition. Then, we develop two types of priors that provide improved generalized Bayes estimators and examine their performance numerically. The proposed priors attain risk reduction by adding scalar shrinkage or column-wise shrinkage to singular value shrinkage. Parallel results for Bayesian predictive densities are also given.

Gaussian processes (GPs) are a highly flexible, nonparametric statistical model that are commonly used to fit nonlinear relationships or account for correlation between observations. However, the computational load of fitting a Gaussian process is $Op{n}^{3}q$ making them infeasible for use on large datasets. To make GPs more feasible for large datasets, this research focuses on the use of minibatching to estimate GP parameters. Specifically, we outline both approximate and exact minibatch Markov chain Monte Carlo algorithms that substantially reduce the computation of fitting a GP by only considering small subsets of the data at a time. We demonstrate and compare this methodology using various simulations and real datasets.

Preferential sampling is a common feature in geostatistics and occurs when the locations are sampled based on information about the phenomena under study. In this case, point pattern models are commonly used as the probability law for the distribution of the locations. However, analytic intractability of the point process likelihood prevents its direct calculation. Many Bayesian (and non-Bayesian) approaches in non-parametric model specifications handle this difficulty with approximations, both to the model and to the computations required for drawing inference. Procedures to approximate the model lead to errors that are sometimes difficult to quantify and can lead to biased inference. This paper presents an approach for performing exact Bayesian inference for this setting without the need for model approximation. A qualitatively minor change on the traditional model is proposed to circumvent the likelihood intractability. This change enables the use of an augmented model strategy. Recent work on Bayesian inference for point pattern models can be adapted to the geostatistics setting and renders computational tractability for exact inference for the proposed methodology. Estimation of model parameters and prediction of the response at unsampled locations can then be obtained from the joint posterior distribution. Simulated studies showed good quality of the proposed model for estimation and prediction in a variety of scenarios. The performance of our approach is illustrated in the analysis of simulated and real datasets and also compares favourably against approximation-based approaches. The paper is concluded with comments regarding extensions and improvements to the proposed methodology.

We propose a novel approach to estimating a multivariate Gaussian precision matrix that relies on decomposing them into a low-rank and a diagonal component. Such decompositions are very popular for modeling large covariance matrices as they admit a latent factor based representation that allows easy inference. The same is however not true for precision matrices due to the lack of computationally convenient representations which restricts inference to low-to-moderate dimensional problems. We address this remarkable gap in the literature by building on a latent variable representation for such decomposition for precision matrices. The construction leads to an efficient Gibbs sampler that scales very well to high-dimensional problems far beyond the limits of the current state-of-the-art. The ability to efficiently explore the full posterior space also allows easy assessment of model uncertainty. Exact zeros in the matrix encoding the underlying conditional independence graph are then determined via a novel posterior false discovery rate control procedure. A near minimax optimal posterior concentration rate for estimating precision matrices is attained by our method under mild regularity assumptions. We evaluate the method’s empirical performance through synthetic experiments and illustrate its practical utility in datasets from two different application domains.

Stick-breaking (SB) processes are often adopted in Bayesian mixture models for generating mixing weights. When covariates influence the sizes of clusters, SB mixtures are particularly convenient as they can leverage their connection to binary regression to ease both the specification of covariate effects and posterior computation. Existing SB models are typically constructed based on continually breaking a single remaining piece of the unit stick. We view this from a dyadic tree perspective in terms of a lopsided bifurcating tree that extends only on one side. We show that two unsavory characteristics of SB models are in fact largely due to this lopsided tree structure. We consider a generalized class of SB models with alternative bifurcating tree structures and examine the influence of the underlying tree topology on the resulting Bayesian analysis in terms of prior assumptions, posterior uncertainty, and computational effectiveness. In particular, we provide evidence that a balanced tree topology, which corresponds to continually breaking all remaining pieces of the unit stick, can resolve or mitigate these undesirable properties of SB models that rely on a lopsided tree.

The use of hierarchical mixture priors with shared atoms has recently flourished in the Bayesian literature for partially exchangeable data. Leveraging on nested levels of mixtures, these models allow the estimation of a two-layered data partition: across groups and across observations. This paper discusses and compares the properties of such modeling strategies when the mixing weights are assigned either a finite-dimensional Dirichlet distribution or a Dirichlet process prior. Based on these considerations, we introduce a novel hierarchical nonparametric prior based on a finite set of shared atoms, a specification that enhances the flexibility of the induced random measures and the availability of fast posterior inference. To support these findings, we analytically derive the induced prior correlation structure and partially exchangeable partition probability function. Additionally, we develop a novel mean-field variational algorithm for posterior inference to boost the applicability of our nested model to large multivariate data. We then assess and compare the performance of the different shared-atom specifications via simulation. We also show that our variational proposal is highly scalable and that the accuracy of the posterior density estimate and the estimated partition is comparable with state-of-the-art Gibbs sampler algorithms. Finally, we apply our model to a real dataset of Spotify’s song features, simultaneously segmenting artists and songs with similar characteristics.

Dirichlet process mixtures are particularly sensitive to the value of the precision parameter controlling the behavior of the latent partition. Randomization of the precision through a prior distribution is a common solution, which leads to more robust inferential procedures. However, existing prior choices do not allow for transparent elicitation, due to the lack of analytical results. We introduce and investigate a novel prior for the Dirichlet process precision, the Stirling-gamma distribution. We study the distributional properties of the induced random partition, with an emphasis on the number of clusters. Our theoretical investigation clarifies the reasons of the improved robustness properties of the proposed prior. Moreover, we show that, under specific choices of its hyperparameters, the Stirling-gamma distribution is conjugate to the random partition of a Dirichlet process. We illustrate with an ecological application the usefulness of our approach for the detection of communities of ant workers.

Computer models are widely used across a range of scientific disciplines to describe various complex physical systems, however to perform full uncertainty quantification we often need to employ emulators. An emulator is a fast statistical construct that mimics the slow to evaluate computer model, and greatly aids the vastly more computationally intensive uncertainty quantification calculations that an important scientific analysis often requires. We examine the problem of emulating computer models that possess multiple, partial discontinuities occurring at known non-linear locations. We introduce the Torn Embedding Non-Stationary Emulation (TENSE) framework, based on carefully designed correlation structures that respect the discontinuities while enabling full exploitation of any smoothness/continuity elsewhere. This leads to a single emulator object that can be updated by all runs simultaneously, and also used for efficient design. This approach avoids having to split the input space into multiple subregions. We apply the TENSE framework to the TNO Challenge II, emulating the OLYMPUS reservoir model, which possesses multiple such discontinuities.

We explore the estimation of generalized additive models using basis expansion in conjunction with Bayesian model selection. Although Bayesian model selection is useful for regression splines, it has traditionally been applied mainly to Gaussian regression owing to the availability of a tractable marginal likelihood. We extend this method to handle an exponential family of distributions by using the Laplace approximation of the likelihood. Although this approach works well with any Gaussian prior distribution, consensus has not been reached on the best prior for nonparametric regression with basis expansions. Our investigation indicates that the classical unit information prior may not be ideal for nonparametric regression. Instead, we find that mixtures of g-priors are more effective. We evaluate various mixtures of g-priors to assess their performance in estimating generalized additive models. Additionally, we compare several priors for knots to determine the most effective strategy. Our simulation studies demonstrate that model selection-based approaches outperform other Bayesian methods.

Meta-analysis aims to generalize results from multiple related statistical analyses through a combined analysis. While the natural outcome of a Bayesian study is a posterior distribution, traditional Bayesian meta-analyses proceed by combining summary statistics (i.e., point-valued estimates) computed from data. In this paper, we develop a framework for combining posterior distributions from multiple related Bayesian studies into a meta-analysis. Importantly, the method is capable of reusing pre-computed posteriors from computationally costly analyses, without needing the implementation details from each study. Besides providing a consensus across studies, the method enables updating the local posteriors post-hoc and therefore refining them by sharing statistical strength between the studies, without rerunning the original analyses. We illustrate the wide applicability of the framework by combining results from likelihood-free Bayesian analyses, which would be difficult to carry out using standard methodology.

Bayesian hypothesis tests leverage posterior probabilities, Bayes factors, or credible intervals to inform data-driven decision making. We propose a framework for power curve approximation with such hypothesis tests. We present a fast approach to explore the approximate sampling distribution of posterior probabilities when the conditions for the Bernstein-von Mises theorem are satisfied. We extend that approach to consider segments of such sampling distributions in a targeted manner for each sample size explored. These sampling distribution segments are used to construct power curves for various types of posterior analyses. Our resulting method for power curve approximation is orders of magnitude faster than conventional power curve estimation for Bayesian hypothesis tests. We also prove the consistency of the corresponding power estimates and sample size recommendations under certain conditions.

The linear varying coefficient models posits a linear relationship between an outcome and covariates in which the covariate effects are modeled as functions of additional effect modifiers. Despite a long history of study and use in statistics and econometrics, state-of-the-art varying coefficient modeling methods cannot accommodate multivariate effect modifiers without imposing restrictive functional form assumptions or involving computationally intensive hyperparameter tuning. In response, we introduce VCBART, which flexibly estimates the covariate effect in a varying coefficient model using Bayesian Additive Regression Trees. With simple default settings, VCBART outperforms existing varying coefficient methods in terms of covariate effect estimation, uncertainty quantification, and outcome prediction. We illustrate the utility of VCBART with two case studies: one examining how the association between later-life cognition and measures of socioeconomic position vary with respect to age and socio-demographics and another estimating how temporal trends in urban crime vary at the neighborhood level. An R package implementing VCBART is available at https://github.com/skdeshpande91/VCBART.

We develop a Bayesian inference method for the parameters of discretely-observed stochastic differential equations (SDEs). Inference is challenging for most SDEs, due to the analytical intractability of the likelihood function. Nevertheless, forward simulation via numerical methods is straightforward, motivating the use of approximate Bayesian computation (ABC). We propose a computationally efficient “data-conditional” simulation scheme for SDEs that is based on lookahead strategies for sequential Monte Carlo (SMC) and particle smoothing using backward simulation. This leads to the simulation of trajectories that are consistent with the observed trajectory, thereby considerably increasing the acceptance rate in an ABC-SMC sampler. As a result, our procedure rapidly guides the parameters towards regions of high posterior density, especially in the first ABC-SMC round. We additionally construct a sequential scheme to learn the ABC summary statistics, by employing an invariant neural network, previously developed for Markov processes, that is incrementally retrained during the run of the ABC-SMC sampler. Our approach achieves accurate inference and is about three times faster (and in some cases even 10 times faster) than standard (forward-only) ABC-SMC. We illustrate our method in five simulation studies, including three examples from the Chan–Karaolyi–Longstaff–Sanders SDE family, a stochastic bi-stable model (Schlögl) that is notoriously challenging for ABC methods, and a two dimensional biochemical reaction network.

A statistical method for the elicitation of priors in Bayesian generalised linear models (GLMs) and extensions is proposed. Probabilistic predictions are elicited from the expert to parametrise a multivariate normal or multivariate t prior distribution for the unknown linear coefficients of the GLM, and an inverse gamma prior for the dispersion parameter if unknown. The elicited predictions condition on defined elicitation scenarios. Dependencies among scenarios are then elicited from the expert by conditioning on hypothetical outcomes. Elicited conditional medians parametrise a canonical vine copula model of dependence that may be truncated for efficiency. The statistical elicitation method permits prior parametrisation of GLMs with alternative choices of design matrices or observation models from the same elicitation session. Extensions of the method apply to multicategorical data, data with bounded support, semi-continuous data with point mass at zero, lognormal and log t models, and count data with overdispersion or zero-inflation. A case study elicits a prior for an extended GLM embedded in a statistical model of overdispersed counts described by a binomial-simplex mixture distribution. The elicited canonical vine model of dependence is found to incorporate substantial information into the prior. The procedures of the statistical elicitation method are implemented in the R package eglm.

Likelihood-based inference in stochastic non-linear dynamical systems, such as those found in chemical reaction networks and biological clock systems, is inherently complex and has largely been limited to small and unrealistically simple systems. Recent advances in analytically tractable approximations to the underlying conditional probability distributions enable long-term dynamics to be accurately modelled, and make the large number of model evaluations required for exact Bayesian inference much more feasible. We propose a new methodology for inference in stochastic non-linear dynamical systems exhibiting oscillatory behaviour that can be applied even if the system involves a large number of variables and unknown parameters. and show the parameters in these models can be realistically estimated from simulated data. We use preliminary analyses based on the Fisher Information Matrix of the model to guide the implementation of Bayesian inference. We show that this parameter sensitivity analysis can predict which parameters are practically identifiable. A parallel tempering algorithm is used to provide the flexibility required to explore the posterior distribution of the model parameters, which often exhibit multi-modal posterior distributions.

We propose a class of nonstationary processes to characterize space- and time-varying directional associations in point-referenced data. We are motivated by spatiotemporal modeling of air pollutants in which local wind patterns are key determinants of the pollutant spread, but information regarding prevailing wind directions may be missing or unreliable. We propose to map a discrete set of wind directions to edges in a sparse directed acyclic graph (DAG), accounting for uncertainty in directional correlation patterns across a domain. The resulting Bag of DAGs processes (BAGs) lead to interpretable nonstationarity and scalability for large data due to sparsity of DAGs in the bag. We outline Bayesian hierarchical models using BAGs and illustrate inferential and performance gains of our methods compared to other state-of-the-art alternatives. We analyze fine particulate matter using high-resolution data from low-cost air quality sensors in California during the 2020 wildfire season. An R package is available on GitHub.

In this paper, we develop a novel spatial variable selection method for scalar on vector-valued image regression in a multi-group setting. Here, ‘vector-valued image’ refers to the imaging datasets that contain vector-valued information at each pixel/voxel location, such as RGB color images, multimodal medical images, DTI imaging, etc. The focus of this work is to identify the spatial locations in the image having an important effect on the scalar outcome measure. Specifically, the overall effect of each voxel is of interest. We thus develop a novel shrinkage prior by soft-thresholding the ${\ell }_2$ norm of a latent multivariate Gaussian process. It allows us to estimate sparse and piecewise-smooth spatially varying vector-valued regression coefficient function. Motivated by the real data, we further develop a double soft-thresholding based framework when there are multiple pre-specified subgroups. For posterior inference, an efficient MCMC algorithm is developed. We compute the posterior contraction rate for parameter estimation and also establish consistency for variable selection of the proposed Bayesian model, assuming that the true regression coefficients are Hölder smooth. Finally, we demonstrate the advantages of the proposed method in simulation studies and further illustrate in an ADNI dataset for modeling MMSE scores based on DTI-based vector-valued imaging markers.

Measurement error in multinomial data is a well-known and well-studied inferential problem that is encountered in many fields, including engineering, biomedical and omics research, ecology, finance, official statistics, and social sciences. Methods developed to accommodate measurement error in multinomial data are typically equipped to handle false negatives or false positives, but not both. We provide a unified framework for accommodating both forms of measurement error using a Bayesian hierarchical approach. We demonstrate the proposed method’s performance on simulated data and apply it to acoustic bat monitoring and official crime data.

For Bayesian hierarchical models represented by directed acyclic graphs, all neighbours of a node provide, potentially conflicting, information on the node. Hence identifying potential conflict between these different information sources can be important for assessing the adequacy of Bayesian hierarchical models. Building on this idea originating from O’Hagan (2003), Gåsemyr and Natvig (2009) constructed node based conflict measures that have been shown to be well calibrated. Sharing the idea of considering potential conflict between separate information sources for each node, Scheel et al. (2011) constructed the graphical diagnostic the local critique plot (lcp). This can be used both to identify conflict at each node, and provide insight into the nature of the conflict, hence being more informative than summary diagnostics. In the present paper we link these two ideas together by constructing a new diagnostic plot, the integrated information contributions local critique plot (iic-lcp), that is supplementary to lcp, but builds on the framework of Gåsemyr and Natvig (2009). It has the advantage over the lcp with the possibility to display curves corresponding to different parameters in the same plot, saving space and easing comparisons, particularly useful for sets of parameters representing exchangeable groups. Notably, we establish a relationship between the geometry of the iic-lcp plot and some of the conflict measures defined in Gåsemyr and Natvig (2009). This indicates how the conflict expressed visually in the graphical plots can be soundly assessed numerically.

Markov chain Monte Carlo (MCMC) algorithms for hidden Markov models often rely on the forward-backward sampler. This makes them computationally slow as the length of the time series increases, motivating the development of sub-sampling-based approaches. These approximate the full posterior by using small random subsequences of the data at each MCMC iteration within stochastic gradient MCMC. In the presence of imbalanced data resulting from rare latent states, subsequences often exclude rare latent state data, leading to inaccurate inference and prediction/detection of rare events. We propose a targeted sub-sampling (TASS) approach that over-samples observations corresponding to rare latent states when calculating the stochastic gradient of parameters associated with them. TASS uses an initial clustering of the data to construct subsequence weights that reduce the variance in gradient estimation. This leads to improved sampling efficiency, in particular in settings where the rare latent states correspond to extreme observations. We demonstrate substantial gains in predictive and inferential accuracy on real and synthetic examples.

In this paper, we propose a scalable Bayesian method for sparse covariance matrix estimation by incorporating a continuous shrinkage prior with a screening procedure. In the first step of the procedure, the off-diagonal elements with small correlations are screened based on their sample correlations. In the second step, the posterior of the covariance with the screened elements fixed at 0 is computed with the beta-mixture prior. The screened elements of the covariance significantly increase the efficiency of the posterior computation. The simulation studies and real data applications show that the proposed method can be used for the high-dimensional problem with the ‘large p, small n’. In some examples in this paper, the proposed method can be computed in a reasonable amount of time, while no other existing Bayesian methods can be. The proposed method has also sound theoretical properties. The screening procedure has the sure screening property and the selection consistency, and the posterior has the optimal minimax or nearly minimax convergence rate under the Frobenius norm.

We consider the problem of aggregating the judgements of a group of experts to form a single prior distribution representing the judgements of the group. We develop a Bayesian hierarchical model to reconcile the judgements of the group of experts based on elicited quantiles for continuous quantities and probabilities for one-off events. Previous Bayesian reconciliation methods have not been used widely, if at all, in contrast to pooling methods and consensus-based approaches. To address this we embed Bayesian reconciliation within the probabilistic Delphi method. The result is to furnish the outcome of the probabilistic Delphi method with a direct probabilistic interpretation, with the resulting prior representing the judgements of the decision maker. We can use the rationales from the Delphi process to group the experts for the hierarchical modelling. We illustrate the approach with applications to studies evaluating erosion in embankment dams and pump failures in a water pumping station, and assess the properties of the approach using the TU Delft database of expert judgement studies. We see that, even using an off-the-shelf implementation of the approach, it out-performs individual experts, equal weighting of experts and the classical method based on the log score.