A general framework for updating belief distributions

A Gibbs Posterior Framework for Fair Clustering

The rise of machine learning-driven decision-making has sparked a growing emphasis on algorithmic fairness. Within the realm of clustering, the notion of balance is utilized as a criterion for attaining fairness, which characterizes a clustering mechanism as fair when the resulting clusters maintain a consistent proportion of observations representing individuals from distinct groups delineated by protected attributes. Building on this idea, the literature has rapidly incorporated a myriad of extensions, devising fair versions of the existing frequentist clustering algorithms, e.g., k-means, k-medioids, etc., that aim at minimizing specific loss functions. These approaches lack uncertainty quantification associated with the optimal clustering configuration and only provide clustering boundaries without quantifying the probabilities associated with each observation belonging to the different clusters. In this article, we intend to offer a novel probabilistic formulation of the fair clustering problem that facilitates valid uncertainty quantification even under mild model misspecifications, without incurring substantial computational overhead. Mixture model-based fair clustering frameworks facilitate automatic uncertainty quantification, but tend to showcase brittleness under model misspecification and involve significant computational challenges. To circumnavigate such issues, we propose a generalized Bayesian fair clustering framework that inherently enjoys decision-theoretic interpretation. Moreover, we devise efficient computational algorithms that crucially leverage techniques from the existing literature on optimal transport and clustering based on loss functions. The gain from the proposed technology is showcased via numerical experiments and real data examples.

Haplotype-based inference of recent effective population size in modern and ancient DNA samples

Individuals sharing recent ancestors are likely to co-inherit large identical-by-descent (IBD) genomic regions. The distribution of these IBD segments in a population may be used to reconstruct past demographic events such as effective population size variation, but accurate IBD detection is difficult in ancient DNA data and in underrepresented populations with limited reference data. In this work, we introduce an accurate method for inferring effective population size variation during the past ~2000 years in both modern and ancient DNA data, called HapNe. HapNe infers recent population size fluctuations using either IBD sharing (HapNe-IBD) or linkage disequilibrium (HapNe-LD), which does not require phasing and can be computed in low coverage data, including data sets with heterogeneous sampling times. HapNe shows improved accuracy in a range of simulated demographic scenarios compared to currently available methods for IBD-based and LD-based inference of recent effective population size, while requiring fewer computational resources. We apply HapNe to several modern populations from the 1,000 Genomes Project, the UK Biobank, the Allen Ancient DNA Resource, and recently published samples from Iron Age Britain, detecting multiple instances of recent effective population size variation across these groups.

Generalized Bayes approach to inverse problems with model misspecification

We propose a general framework for obtaining probabilistic solutions to PDE-based inverse problems. Bayesian methods are attractive for uncertainty quantification but assume knowledge of the likelihood model or data generation process. This assumption is difficult to justify in many inverse problems, where the specification of the data generation process is not obvious. We adopt a Gibbs posterior framework that directly posits a regularized variational problem on the space of probability distributions of the parameter. We propose a novel model comparison framework that evaluates the optimality of a given loss based on its "predictive performance". We provide cross-validation procedures to calibrate the regularization parameter of the variational objective and compare multiple loss functions. Some novel theoretical properties of Gibbs posteriors are also presented. We illustrate the utility of our framework via a simulated example, motivated by dispersion-based wave models used to characterize arterial vessels in ultrasound vibrometry.

Semiparametric Bayesian inference for optimal dynamic treatment regimes via dynamic marginal structural models

Considerable statistical work done on dynamic treatment regimes (DTRs) is in the frequentist paradigm, but Bayesian methods may have much to offer in this setting as they allow for the appropriate representation and propagation of uncertainty, including at the individual level. In this work, we extend the use of recently developed Bayesian methods for Marginal Structural Models to arrive at inference of DTRs. We do this (i) by linking the observational world with a world in which all patients are randomized to a DTR, thereby allowing for causal inference and then (ii) by maximizing a posterior predictive utility, where the posterior distribution has been obtained from nonparametric prior assumptions on the observational world data-generating process. Our approach relies on Bayesian semiparametric inference, where inference about a finite-dimensional parameter is made all while working within an infinite-dimensional space of distributions. We further study Bayesian inference of DTRs in the double robust setting by using posterior predictive inference and the nonparametric Bayesian bootstrap. The proposed methods allow for uncertainty quantification at the individual level, thereby enabling personalized decision-making. We examine the performance of these methods via simulation and demonstrate their utility by exploring whether to adapt HIV therapy to a measure of patients' liver health, in order to minimize liver scarring.

Bayesian cluster analysis

Bayesian cluster analysis offers substantial benefits over algorithmic approaches by providing not only point estimates but also uncertainty in the clustering structure and patterns within each cluster. An overview of Bayesian cluster analysis is provided, including both model-based and loss-based approaches, along with a discussion on the importance of the kernel or loss selected and prior specification. Advantages are demonstrated in an application to cluster cells and discover latent cell types in single-cell RNA sequencing data to study embryonic cellular development. Lastly, we focus on the ongoing debate between finite and infinite mixtures in a model-based approach and robustness to model misspecification. While much of the debate and asymptotic theory focuses on the marginal posterior of the number of clusters, we empirically show that quite a different behaviour is obtained when estimating the full clustering structure. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Deep bootstrap for Bayesian inference

For a Bayesian, the task to define the likelihood can be as perplexing as the task to define the prior. We focus on situations when the parameter of interest has been emancipated from the likelihood and is linked to data directly through a loss function. We survey existing work on both Bayesian parametric inference with Gibbs posteriors and Bayesian non-parametric inference. We then highlight recent bootstrap computational approaches to approximating loss-driven posteriors. In particular, we focus on implicit bootstrap distributions defined through an underlying push-forward mapping. We investigate independent, identically distributed (iid) samplers from approximate posteriors that pass random bootstrap weights through a trained generative network. After training the deep-learning mapping, the simulation cost of such iid samplers is negligible. We compare the performance of these deep bootstrap samplers with exact bootstrap as well as MCMC on several examples (including support vector machines or quantile regression). We also provide theoretical insights into bootstrap posteriors by drawing upon connections to model mis-specification. This article is part of the theme issue 'Bayesian inference: challenges, perspectives, and prospects'.

Quantifying the rebound of influenza epidemics after the adjustment of zero-COVID policy in China

The coexistence of coronavirus disease 2019 (COVID-19) and seasonal influenza epidemics has become a potential threat to human health, particularly in China in the oncoming season. However, with the relaxation of nonpharmaceutical interventions (NPIs) during the COVID-19 pandemic, the rebound extent of the influenza activities is still poorly understood. In this study, we constructed a susceptible-vaccinated-infectious-recovered-susceptible (SVIRS) model to simulate influenza transmission and calibrated it using influenza surveillance data from 2018 to 2022. We projected the influenza transmission over the next 3 years using the SVIRS model. We observed that, in epidemiological year 2021-2022, the reproduction numbers of influenza in southern and northern China were reduced by 64.0 and 34.5%, respectively, compared with those before the pandemic. The percentage of people susceptible to influenza virus increased by 138.6 and 57.3% in southern and northern China by October 1, 2022, respectively. After relaxing NPIs, the potential accumulation of susceptibility to influenza infection may lead to a large-scale influenza outbreak in the year 2022-2023, the scale of which may be affected by the intensity of the NPIs. And later relaxation of NPIs in the year 2023 would not lead to much larger rebound of influenza activities in the year 2023-2024. To control the influenza epidemic to the prepandemic level after relaxing NPIs, the influenza vaccination rates in southern and northern China should increase to 53.8 and 33.8%, respectively. Vaccination for influenza should be advocated to reduce the potential reemergence of the influenza epidemic in the next few years.

Direct Incorporation of Expert Opinion into Parametric Survival Models to Inform Survival Extrapolation

BACKGROUND

In decision modeling with time-to-event data, there are a variety of parametric models that can be used to extrapolate the survival function. Each model implies a different hazard function, and in situations in which there is moderate censoring, this can result in quite different survival projections. External information such as expert opinion on long-term survival can more accurately characterize the uncertainty in these extrapolations.

OBJECTIVE

We present a general and easily implementable approach to incorporate various types of expert opinions into parametric survival models, focusing on opinions about survival at various landmark time points.

METHODS

Expert opinion is incorporated into parametric survival models using Bayesian and frequentist approaches. In the Bayesian method, expert opinion is included through a loss function and in the frequentist approach by penalizing the likelihood function, although in both cases the core approach is the same. The issue of aggregating multiple expert opinions is also considered.

RESULTS

We apply this method to data from a leukemia trial and use previously elicited expert opinion on survival probabilities for that particular trial population at years 4 and 5 to inform our analysis. We take a robust approach to modeling expert opinion by using pooled distributions and fit a broad class of parametric models to the data. We also assess statistical goodness of fit of the models to both the observed data and expert opinion.

CONCLUSIONS

Expert opinions can be implemented in a straightforward manner using this novel approach; however, more work is required on the correct elicitation of these quantities.

HIGHLIGHTS

Presentation of a novel and open-source method to incorporate expert opinion into decision modeling.Extends upon earlier work in that expert opinion can be incorporated into a wide range of parametric models.Provides methodological guidance for directly including expert opinion in decision modeling, which is a research focus area in NICE TSD 21.¹.

From Bilinear Regression to Inductive Matrix Completion: A Quasi-Bayesian Analysis

In this paper, we study the problem of bilinear regression, a type of statistical modeling that deals with multiple variables and multiple responses. One of the main difficulties that arise in this problem is the presence of missing data in the response matrix, a problem known as inductive matrix completion. To address these issues, we propose a novel approach that combines elements of Bayesian statistics with a quasi-likelihood method. Our proposed method starts by addressing the problem of bilinear regression using a quasi-Bayesian approach. The quasi-likelihood method that we employ in this step allows us to handle the complex relationships between the variables in a more robust way. Next, we adapt our approach to the context of inductive matrix completion. We make use of a low-rankness assumption and leverage the powerful PAC-Bayes bound technique to provide statistical properties for our proposed estimators and for the quasi-posteriors. To compute the estimators, we propose a Langevin Monte Carlo method to obtain approximate solutions to the problem of inductive matrix completion in a computationally efficient manner. To demonstrate the effectiveness of our proposed methods, we conduct a series of numerical studies. These studies allow us to evaluate the performance of our estimators under different conditions and provide a clear illustration of the strengths and limitations of our approach.

Generalized Geographically Weighted Regression Model within a Modularized Bayesian Framework

Geographically weighted regression (GWR) models handle geographical dependence through a spatially varying coefficient model and have been widely used in applied science, but its general Bayesian extension is unclear because it involves a weighted log-likelihood which does not imply a probability distribution on data. We present a Bayesian GWR model and show that its essence is dealing with partial misspecification of the model. Current modularized Bayesian inference models accommodate partial misspecification from a single component of the model. We extend these models to handle partial misspecification in more than one component of the model, as required for our Bayesian GWR model. Information from the various spatial locations is manipulated via a geographically weighted kernel and the optimal manipulation is chosen according to a Kullback-Leibler (KL) divergence. We justify the model via an information risk minimization approach and show the consistency of the proposed estimator in terms of a geographically weighted KL divergence.

Tailored Bayes: a risk modeling framework under unequal misclassification costs

Risk prediction models are a crucial tool in healthcare. Risk prediction models with a binary outcome (i.e., binary classification models) are often constructed using methodology which assumes the costs of different classification errors are equal. In many healthcare applications, this assumption is not valid, and the differences between misclassification costs can be quite large. For instance, in a diagnostic setting, the cost of misdiagnosing a person with a life-threatening disease as healthy may be larger than the cost of misdiagnosing a healthy person as a patient. In this article, we present Tailored Bayes (TB), a novel Bayesian inference framework which "tailors" model fitting to optimize predictive performance with respect to unbalanced misclassification costs. We use simulation studies to showcase when TB is expected to outperform standard Bayesian methods in the context of logistic regression. We then apply TB to three real-world applications, a cardiac surgery, a breast cancer prognostication task, and a breast cancer tumor classification task and demonstrate the improvement in predictive performance over standard methods.

BAYESIAN FUNCTIONAL REGISTRATION OF FMRI ACTIVATION MAPS

Functional magnetic resonance imaging (fMRI) has provided invaluable insight into our understanding of human behavior. However, large inter-individual differences in both brain anatomy and functional localization after anatomical alignment remain a major limitation in conducting group analyses and performing population level inference. This paper addresses this problem by developing and validating a new computational technique for reducing misalignment across individuals in functional brain systems by spatially transforming each subjects functional data to a common reference map. Our proposed Bayesian functional registration approach allows us to assess differences in brain function across subjects and individual differences in activation topology. It combines intensity-based and feature-based information into an integrated framework, and allows inference to be performed on the transformation via the posterior samples. We evaluate the method in a simulation study and apply it to data from a study of thermal pain. We find that the proposed approach provides increased sensitivity for group-level inference.

An efficient adaptive MCMC algorithm for Pseudo-Bayesian quantum tomography

We revisit the Pseudo-Bayesian approach to the problem of estimating density matrix in quantum state tomography in this paper. Pseudo-Bayesian inference has been shown to offer a powerful paradigm for quantum tomography with attractive theoretical and empirical results. However, the computation of (Pseudo-)Bayesian estimators, due to sampling from complex and high-dimensional distribution, pose significant challenges that hamper their usages in practical settings. To overcome this problem, we present an efficient adaptive MCMC sampling method for the Pseudo-Bayesian estimator by exploring an adaptive proposal scheme together with subsampling method. We show in simulations that our approach is substantially computationally faster than the previous implementation by at least two orders of magnitude which is significant for practical quantum tomography.

Consensus clustering for Bayesian mixture models

BACKGROUND

Cluster analysis is an integral part of precision medicine and systems biology, used to define groups of patients or biomolecules. Consensus clustering is an ensemble approach that is widely used in these areas, which combines the output from multiple runs of a non-deterministic clustering algorithm. Here we consider the application of consensus clustering to a broad class of heuristic clustering algorithms that can be derived from Bayesian mixture models (and extensions thereof) by adopting an early stopping criterion when performing sampling-based inference for these models. While the resulting approach is non-Bayesian, it inherits the usual benefits of consensus clustering, particularly in terms of computational scalability and providing assessments of clustering stability/robustness.

RESULTS

In simulation studies, we show that our approach can successfully uncover the target clustering structure, while also exploring different plausible clusterings of the data. We show that, when a parallel computation environment is available, our approach offers significant reductions in runtime compared to performing sampling-based Bayesian inference for the underlying model, while retaining many of the practical benefits of the Bayesian approach, such as exploring different numbers of clusters. We propose a heuristic to decide upon ensemble size and the early stopping criterion, and then apply consensus clustering to a clustering algorithm derived from a Bayesian integrative clustering method. We use the resulting approach to perform an integrative analysis of three 'omics datasets for budding yeast and find clusters of co-expressed genes with shared regulatory proteins. We validate these clusters using data external to the analysis.

CONCLUSTIONS

Our approach can be used as a wrapper for essentially any existing sampling-based Bayesian clustering implementation, and enables meaningful clustering analyses to be performed using such implementations, even when computational Bayesian inference is not feasible, e.g. due to poor exploration of the target density (often as a result of increasing numbers of features) or a limited computational budget that does not along sufficient samples to drawn from a single chain. This enables researchers to straightforwardly extend the applicability of existing software to much larger datasets, including implementations of sophisticated models such as those that jointly model multiple datasets.

A generalized likelihood-based Bayesian approach for scalable joint regression and covariance selection in high dimensions

The paper addresses joint sparsity selection in the regression coefficient matrix and the error precision (inverse covariance) matrix for high-dimensional multivariate regression models in the Bayesian paradigm. The selected sparsity patterns are crucial to help understand the network of relationships between the predictor and response variables, as well as the conditional relationships among the latter. While Bayesian methods have the advantage of providing natural uncertainty quantification through posterior inclusion probabilities and credible intervals, current Bayesian approaches either restrict to specific sub-classes of sparsity patterns and/or are not scalable to settings with hundreds of responses and predictors. Bayesian approaches which only focus on estimating the posterior mode are scalable, but do not generate samples from the posterior distribution for uncertainty quantification. Using a bi-convex regression based generalized likelihood and spike-and-slab priors, we develop an algorithm called Joint Regression Network Selector (JRNS) for joint regression and covariance selection which (a) can accommodate general sparsity patterns, (b) provides posterior samples for uncertainty quantification, and (c) is scalable and orders of magnitude faster than the state-of-the-art Bayesian approaches providing uncertainty quantification. We demonstrate the statistical and computational efficacy of the proposed approach on synthetic data and through the analysis of selected cancer data sets. We also establish high-dimensional posterior consistency for one of the developed algorithms.

A nonparametric Bayesian method for dose finding in drug combinations cancer trials

We propose an adaptive design for early-phase drug-combination cancer trials with the goal of estimating the maximum tolerated dose (MTD). A nonparametric Bayesian model, using beta priors truncated to the set of partially ordered dose combinations, is used to describe the probability of dose limiting toxicity (DLT). Dose allocation between successive cohorts of patients is estimated using a modified continual reassessment scheme. The updated probabilities of DLT are calculated with a Gibbs sampler that employs a weighting mechanism to calibrate the influence of data vs the prior. At the end of the trial, we recommend one or more dose combinations as the MTD based on our proposed algorithm. We apply our method to a Phase I clinical trial of CB-839 and Gemcitabine that motivated this nonparametric design. The design operating characteristics indicate that our method is comparable with existing methods.

Bayes Posterior Convergence for Loss Functions via Almost Additive Thermodynamic Formalism

Statistical inference can be seen as information processing involving input information and output information that updates belief about some unknown parameters. We consider the Bayesian framework for making inferences about dynamical systems from ergodic observations, where the Bayesian procedure is based on the Gibbs posterior inference, a decision process generalization of standard Bayesian inference (see [7, 37]) where the likelihood is replaced by the exponential of a loss function. In the case of direct observation and almost-additive loss functions, we prove an exponential convergence of the a posteriori measures to a limit measure. Our estimates on the Bayes posterior convergence for direct observation are related and extend those in [47] to a context where loss functions are almost-additive. Our approach makes use of non-additive thermodynamic formalism and large deviation properties [39, 40, 57] instead of joinings.

Bayesian variable selection with a pleiotropic loss function in Mendelian randomization

Mendelian randomization is the use of genetic variants as instruments to assess the existence of a causal relationship between a risk factor and an outcome. A Mendelian randomization analysis requires a set of genetic variants that are strongly associated with the risk factor and only associated with the outcome through their effect on the risk factor. We describe a novel variable selection algorithm for Mendelian randomization that can identify sets of genetic variants which are suitable in both these respects. Our algorithm is applicable in the context of two-sample summary-data Mendelian randomization and employs a recently proposed theoretical extension of the traditional Bayesian statistics framework, including a loss function to penalize genetic variants that exhibit pleiotropic effects. The algorithm offers robust inference through the use of model averaging, as we illustrate by running it on a range of simulation scenarios and comparing it against established pleiotropy-robust Mendelian randomization methods. In a real-data application, we study the effect of systolic and diastolic blood pressure on the risk of suffering from coronary heart disease (CHD). Based on a recent large-scale GWAS for blood pressure, we use 395 genetic variants for systolic and 391 variants for diastolic blood pressure. Both traits are shown to have significant risk-increasing effects on CHD risk.

A Discrete Density Approach to Bayesian Quantile and Expectile Regression with Discrete Responses

For decades, regression models beyond the mean for continuous responses have attracted great attention in the literature. These models typically include quantile regression and expectile regression. But there is little research on these regression models for discrete responses, particularly from a Bayesian perspective. By forming the likelihood function based on suitable discrete probability mass functions, this paper introduces a discrete density approach for Bayesian inference of these regression models with discrete responses. Bayesian quantile regression for discrete responses is first developed, and then this method is extended to Bayesian expectile regression for discrete responses. The posterior distribution under this approach is shown not only coherent irrespective of the true distribution of the response, but also proper with regarding to improper priors for the unknown model parameters. The performance of the method is evaluated via extensive Monte Carlo simulation studies and one real data analysis.

A Comparison of Penalized Maximum Likelihood Estimation and Markov Chain Monte Carlo Techniques for Estimating Confirmatory Factor Analysis Models With Small Sample Sizes

With small to modest sample sizes and complex models, maximum likelihood (ML) estimation of confirmatory factor analysis (CFA) models can show serious estimation problems such as non-convergence or parameter estimates outside the admissible parameter space. In this article, we distinguish different Bayesian estimators that can be used to stabilize the parameter estimates of a CFA: the mode of the joint posterior distribution that is obtained from penalized maximum likelihood (PML) estimation, and the mean (EAP), median (Med), or mode (MAP) of the marginal posterior distribution that are calculated by using Markov Chain Monte Carlo (MCMC) methods. In two simulation studies, we evaluated the performance of the Bayesian estimators from a frequentist point of view. The results show that the EAP produced more accurate estimates of the latent correlation in many conditions and outperformed the other Bayesian estimators in terms of root mean squared error (RMSE). We also argue that it is often advantageous to choose a parameterization in which the main parameters of interest are bounded, and we suggest the four-parameter beta distribution as a prior distribution for loadings and correlations. Using simulated data, we show that selecting weakly informative four-parameter beta priors can further stabilize parameter estimates, even in cases when the priors were mildly misspecified. Finally, we derive recommendations and propose directions for further research.

Information-Theoretic Generalization Bounds for Meta-Learning and Applications

Meta-learning, or “learning to learn”, refers to techniques that infer an inductive bias from data corresponding to multiple related tasks with the goal of improving the sample efficiency for new, previously unobserved, tasks. A key performance measure for meta-learning is the meta-generalization gap, that is, the difference between the average loss measured on the meta-training data and on a new, randomly selected task. This paper presents novel information-theoretic upper bounds on the meta-generalization gap. Two broad classes of meta-learning algorithms are considered that use either separate within-task training and test sets, like model agnostic meta-learning (MAML), or joint within-task training and test sets, like reptile. Extending the existing work for conventional learning, an upper bound on the meta-generalization gap is derived for the former class that depends on the mutual information (MI) between the output of the meta-learning algorithm and its input meta-training data. For the latter, the derived bound includes an additional MI between the output of the per-task learning procedure and corresponding data set to capture within-task uncertainty. Tighter bounds are then developed for the two classes via novel individual task MI (ITMI) bounds. Applications of the derived bounds are finally discussed, including a broad class of noisy iterative algorithms for meta-learning.

On Default Priors for Robust Bayesian Estimation with Divergences

This paper presents objective priors for robust Bayesian estimation against outliers based on divergences. The minimum γ-divergence estimator is well-known to work well in estimation against heavy contamination. The robust Bayesian methods by using quasi-posterior distributions based on divergences have been also proposed in recent years. In the objective Bayesian framework, the selection of default prior distributions under such quasi-posterior distributions is an important problem. In this study, we provide some properties of reference and moment matching priors under the quasi-posterior distribution based on the γ-divergence. In particular, we show that the proposed priors are approximately robust under the condition on the contamination distribution without assuming any conditions on the contamination ratio. Some simulation studies are also presented.

Approximate Bayesian Inference

This is the Editorial article summarizing the scope of the Special Issue: Approximate Bayesian Inference.

Statistical decision theory and multiscale analyses of human brain data

BACKGROUND

In the era of Big Data, large scale electrophysiological data from animal and human studies are abundant. These data contain information at multiple spatiotemporal scales. However, current approaches for the analysis of electrophysiological data often focus on a single spatiotemporal scale only.

NEW METHOD

We discuss a multiscale approach for the analysis of electrophysiological data. This is based on combining neural models that describe brain data at different scales. It allows us to make laminar-specific inferences about neurobiological properties of cortical sources using non invasive human electrophysiology data.

RESULTS

We provide a mathematical proof of this approach using statistical decision theory. We also consider its extensions to brain imaging studies including data from the same subjects performing different tasks. As an illustration, we show that changes in gamma oscillations between different people might originate from differences in recurrent connection strengths of inhibitory interneurons in layers 5/6.

COMPARISON WITH EXISTING METHODS

This is a new approach that follows up on our recent work. It is different from other approaches where the scale of spatiotemporal dynamics is fixed.

CONCLUSIONS

We discuss a multiscale approach for the analysis of human MEG data. This uses a neural mass model that includes constraints informed by a compartmental model. This has two advantages. First, it allows us to find differences in cortical laminar dynamics and understand neurobiological properties like neuromodulation, excitation to inhibition balance etc. using non invasive data. Second, it allows us to validate macroscale models by exploiting animal data.

Robust Bayesian Regression with Synthetic Posterior Distributions

Although linear regression models are fundamental tools in statistical science, the estimation results can be sensitive to outliers. While several robust methods have been proposed in frequentist frameworks, statistical inference is not necessarily straightforward. We here propose a Bayesian approach to robust inference on linear regression models using synthetic posterior distributions based on γ-divergence, which enables us to naturally assess the uncertainty of the estimation through the posterior distribution. We also consider the use of shrinkage priors for the regression coefficients to carry out robust Bayesian variable selection and estimation simultaneously. We develop an efficient posterior computation algorithm by adopting the Bayesian bootstrap within Gibbs sampling. The performance of the proposed method is illustrated through simulation studies and applications to famous datasets.

Nonparametric targeted Bayesian estimation of class proportions in unlabeled data

We introduce a novel Bayesian estimator for the class proportion in an unlabeled dataset, based on the targeted learning framework. The procedure requires the specification of a prior (and outputs a posterior) only for the target of inference, and yields a tightly concentrated posterior. When the scientific question can be characterized by a low-dimensional parameter functional, this focus on target prior and posterior distributions perfectly aligns with Bayesian subjectivism. We prove a Bernstein-von Mises-type result for our proposed Bayesian procedure, which guarantees that the posterior distribution converges to the distribution of an efficient, asymptotically linear estimator. In particular, the posterior is Gaussian, doubly robust, and efficient in the limit, under the only assumption that certain nuisance parameters are estimated at slower-than-parametric rates. We perform numerical studies illustrating the frequentist properties of the method. We also illustrate their use in a motivating application to estimate the proportion of embolic strokes of undetermined source arising from occult cardiac sources or large-artery atherosclerotic lesions. Though we focus on the motivating example of the proportion of cases in an unlabeled dataset, the procedure is general and can be adapted to estimate any pathwise differentiable parameter in a non-parametric model.

Inference and computation with generalized additive models and their extensions

Regression models in which a response variable is related to smooth functions of some predictor variables are popular as a result of their appealing balance between flexibility and interpretability. Since the original generalized additive models of Hastie and Tibshirani (Generalized additive models. Chapman & Hall, Boca Raton, 1990) numerous model extensions have been proposed, and a variety of practically useful computational strategies have emerged. This paper provides an overview of some widely applicable frameworks for this type of modelling, emphasizing the similarities between the different approaches, and the equivalence of smoothing, Gaussian latent process models and Gaussian random effects. The focus is particularly on Bayes empirical smoother theory, fully Bayesian inference via stochastic simulation or integrated nested Laplace approximation and boosting.