Nonparametric Bayesian Analysis Research Articles

Single-molecule tracking with nonparametric Bayesian analysis reveals that histone modification protein complexes form specifically at heterochromatin sites.

Eukaryotic cells can establish distinct heritable phenotypic states with epigenetic modifications. Histone H3K9 methylation (H3K9me) is a conserved epigenetic modification that is associated with transcriptional silencing and heterochromatin formation. The diverse functions of heterochromatin depend on the evolutionarily conserved HP1 family of proteins that recognize and bind to H3K9me nucleosomes. Schizosaccharomyces pombe (fission yeast) have two HP1 homologs, Swi6 and Chp2, which play indispensable roles in regulating heterochromatin structure. Swi6 simultaneously binds to H3K9me and oligomerizes, which enables the spreading of heterochromatin and transcriptional silencing. On the other hand, Chp2 binds to H3K9me and recruits the Nucleosome Remodeling and Deacetylase complex, SHREC, through interaction with the histone remodeler Mit1. The putative H3K9 demethylase Epe1 interacts with Swi6 while the histone deacetylase Clr3 is a part of SHREC, and Epe1 and Clr3 are shown to have opposing effects on nucleosome turnover and heterochromatin formation. Structural biology and in vitro biochemistry assays have revealed the interactions between these histone modification proteins involved in the heterochromatin formation, however, there has not yet been any in vivo measurement of the interaction between these proteins. Here, we use live-cell single-particle tracking (SPT) and super-resolution microscopy to investigate the in vivo dynamics and spatial organization of histone modification proteins. With nonparametric Bayesian analysis of SPT trajectories, we measure distinct biophysical mobility states and map these states to the potential in vivo biochemical role. We propose a new mechanism for histone modification proteins in which the heterochromatin site is the primary location for protein complex formation, and H3K9me is a required substrate for the interaction.

Toward merging image analysis and gene network discovery from fluorescent in situ hybridization data.

Cellular response to extracellular stresses can now be quantitatively monitored using fluorescent in situ hybridization (FISH). Deducing a cell's transcriptional response mechanism from FISH data requires counting individual RNA transcripts within cells and subsequently enumerating the modes of gene expression (termed gene states). However, when the number of gene states are unknown standard likelihood based parametric inference methods are insufficient, as they require pre-specifying the number of gene states. Here we explore a nonparametric progression of these likelihood-based approaches to deduce the number of gene states alongside relevant rates. If time allows, we will also discuss our method for providing the initial snap-shot data for gene network inference (which requires quantifying diffraction-limited RNA spots in crowded environments): a Bayesian nonparametric image analysis tool which attempts to realize the theoretical maximum number of RNA transcripts identifiable from multiplexed FISH data.

Theoretical challenges in single molecule.

Biological processes are directed by molecular actors conspiring, often in small numbers, to achieve tasks relevant to life. Yet these actors are not directly observable. Within the fluorescence paradigm, multiple layers of stochasticity lie between what we care about (molecular actor behavior) and what we observe (detector counts). These hierarchically organized layers include, but are not limited to, incomplete labeling, label photo-physics, optical aberrations, and detector noise. As a direct consequence, it is not possible to speak of discovering unique models of molecular behavior from data. Indeed many models are consistent with the data and our goal as theorists is to assign probabilities to candidate models based on observation. Here we give an overview of hierarchical Bayesian inference models and MCMC strategies for single molecule experiments with an emphasis on nonparametric Bayesian analysis of single photon smFRET highlighting a sequence of three papers to appear in Biophysical Reports.

Comorbidity clusters associated with newly treated type 2 diabetes mellitus: a Bayesian nonparametric analysis

Type 2 diabetes mellitus (T2DM) is associated with the development of chronic comorbidities, which can lead to high drug utilization and adverse events. We aimed to identify common comorbidity clusters and explore the progression over time in newly treated T2DM patients. The IQVIA Medical Research Data incorporating data from THIN, a Cegedim database of anonymized electronic health records, was used to identify all patients with a first-ever prescription for a non-insulin antidiabetic drug (NIAD) between January 2006 and December 2019. We selected 58 chronic comorbidities of interest and used Bayesian nonparametric models to identify disease clusters and model their progression over time. Among the 175,383 eligible T2DM patients, we identified the 20 most frequent comorbidity clusters, which were comprised of 14 latent features (LFs). Each LF was associated with a primary disease (e.g., 98% of patients in cluster 2, characterized by LF2, had congestive heart failure [CHF]). The presence of certain LFs increased the probability of having another LF active. For example, LF2 (CHF) frequently appeared with LFs related to chronic kidney disease (CKD). Over time, the clusters associated with cardiovascular diseases, such as CHF, progressed rapidly. Moreover, the onset of certain diseases led to further complications. Our models identified established T2DM complications and previously unknown connections, thus, highlighting the potential for Bayesian nonparametric models to characterize complex comorbidity patterns.

A nonparametric Bayesian analysis for meningococcal disease counts based on integer-valued threshold time series models

To better describe the characteristics of time series of counts showing over-dispersion, asymmetry and structural change features, this paper considers a class of random coefficient integer-valued threshold autoregressive processes that properly capture flexible asymmetric and nonlinear responses without assuming the distributions for the errors. By the Taylor expansion, a normally distributed working likelihood is obtained and the Bayesian empirical likelihood inference is conducted for the model parameters. Through the normalized approximation of the nonparametric likelihood, the originally complicated Bayesian calculation has been greatly simplified. Finally, the proposed method is verified via some simulations and an empirical analysis of the meningococcal disease data set in Germany.

NONPARAMETRIC BAYES ANALYSIS OF THE SHARP AND FUZZY REGRESSION DISCONTINUITY DESIGNS

We develop a nonparametric Bayesian analysis of regression discontinuity (RD) designs, allowing for covariates, in which we model and estimate the unknown functions of the forcing variable by basis expansion methods. In a departure from current methods, we use the entire data on the forcing variable, but we emphasize the data near the threshold by placing some knots at and near the threshold, a technique we refer to as soft-windowing. To handle the nonequally spaced knots that emerge from soft-windowing, we construct a prior on the spline coefficients, from a second-order Ornstein–Uhlenbeck process, which is hyperparameter light, and satisfies the Kullback–Leibler support property. In the fuzzy RD design, we explain the divergence between the treatment implied by the forcing variable, and the actual intake, by a discrete confounder variable, taking three values, complier, never-taker, and always-taker, and a model with four potential outcomes. Choice of the soft-window, and the number of knots, is determined by marginal likelihoods, computed by the method of Chib [Journal of the American Statistical Association, 1995, 90, 1313–1321] as a by-product of the Markov chain Monte Carlo (MCMC)-based estimation. Importantly, in each case, we allow for covariates, incorporated nonparametrically by additive natural cubic splines. The potential outcome error distributions are modeled as student-t, with an extension to Dirichlet process mixtures. We derive the large sample posterior consistency, and posterior contraction rate, of the RD average treatment effect (ATE) (in the sharp case) and RD ATE for compliers (in the fuzzy case), as the number of basis parameters increases with sample size. The excellent performance of the methods is documented in simulation experiments, and in an application to educational attainment of women from Meyersson [Econometrica, 2014, 82, 229–269].

Bayesian nonparametric analysis for the detection of spikes in noisy calcium imaging data.

Recent advancements in miniaturized fluorescence microscopy have made it possible to investigate neuronal responses to external stimuli in awake behaving animals through the analysis of intracellular calcium signals. An ongoing challenge is deconvolving the temporal signals to extract the spike trains from the noisy calcium signals' time series. In this article, we propose a nested Bayesian finite mixture specification that allows the estimation of spiking activity and, simultaneously, reconstructing the distributions of the calcium transient spikes' amplitudes under different experimental conditions. The proposed model leverages two nested layers of random discrete mixture priors to borrow information between experiments and discover similarities in the distributional patterns of neuronal responses to different stimuli. Furthermore, the spikes' intensity values are also clustered within and between experimental conditions to determine the existence of common (recurring) response amplitudes. Simulation studies and the analysis of a dataset from the Allen Brain Observatory show the effectiveness of the method in clustering and detecting neuronal activities.

Gibbs posterior convergence and the thermodynamic formalism

In this paper we consider the posterior consistency of Bayesian inference procedures when the family of models consists of appropriate stochastic processes. Specifically, we suppose that one observes an unknown ergodic process and one has access to a family of models consisting of dependent processes. In this context, we consider Gibbs posterior inference, which is a loss-based generalization of standard Bayesian inference. Our main results characterize the asymptotic behavior of the Gibbs posterior distributions on the space of models. Furthermore, we show that in the case of properly specified models our convergence results may be used to establish posterior consistency. Our model processes are defined via the thermodynamic formalism for dynamical systems, and they allow for a large degree of dependence, including both Markov chains of unbounded orders and processes that are not Markov of any order. This work establishes close connections between Gibbs posterior inference and the thermodynamic formalism for dynamical systems, which we hope will lead to new questions and results in both nonparametric Bayesian analysis and the thermodynamic formalism.

Bayesian nonparametric analysis of restricted mean survival time.

The restricted mean survival time (RMST) evaluates the expectation of survival time truncated by a prespecified time point, because the mean survival time in the presence of censoring is typically not estimable. The frequentist inference procedure for RMST has been widely advocated for comparison of two survival curves, while research from the Bayesian perspective is rather limited. For the RMST of both right- and interval-censored data, we propose Bayesian nonparametric estimation and inference procedures. By assigning a mixture of Dirichlet processes (MDP) prior to the distribution function, we can estimate the posterior distribution of RMST. We also explore another Bayesian nonparametric approach using the Dirichlet process mixture model and make comparisons with the frequentist nonparametric method. Simulation studies demonstrate that the Bayesian nonparametric RMST under diffuse MDP priors leads to robust estimation and under informative priors it can incorporate prior knowledge into the nonparametric estimator. Analysis of real trial examples demonstrates the flexibility and interpretability of the Bayesian nonparametric RMST for both right- and interval-censored data.

Bernstein-von Mises theorem for the Pitman-Yor process of nonnegative type

The Pitman-Yor process is a random probability distribution, that can be used as a prior distribution in a nonparametric Bayesian analysis. The process is of species sampling type and generates discrete distributions, which yield of the order nσ different values (“species”) in a random sample of size n, if the type σ is positive. Thus this type parameter can be set to target true distributions of various levels of discreteness, making the Pitman-Yor process an interesting prior in this case. It was previously shown that the resulting posterior distribution is consistent if and only if the true distribution of the data is discrete. In this paper we derive the distributional limit of the posterior distribution, in the form of a (corrected) Bernstein-von Mises theorem, which previously was known only in the continuous, inconsistent case. It turns out that the Pitman-Yor posterior distribution has good behaviour if the true distribution of the data is discrete with atoms that decrease not too slowly. Credible sets derived from the posterior distribution provide valid frequentist confidence sets in this case. For a general discrete distribution, the posterior distribution, although consistent, may contain a bias which does not converge to zero at the n rate and invalidates posterior inference. We propose a bias correction that solves this problem. We also consider the effect of estimating the type parameter from the data, both by empirical Bayes and full Bayes methods. In a small simulation study we illustrate that without bias correction the coverage of credible sets can be arbitrarily low, also for some discrete distributions.

A Bayesian Nonparametric Analysis of the 2003 Outbreak of Highly Pathogenic Avian Influenza in the Netherlands

Abstract Infectious diseases on farms pose both public and animal health risks, so understanding how they spread between farms is crucial for developing disease control strategies to prevent future outbreaks. We develop novel Bayesian nonparametric methodology to fit spatial stochastic transmission models in which the infection rate between any two farms is a function that depends on the distance between them, but without assuming a specified parametric form. Making nonparametric inference in this context is challenging since the likelihood function of the observed data is intractable because the underlying transmission process is unobserved. We adopt a fully Bayesian approach by assigning a transformed Gaussian process prior distribution to the infection rate function, and then develop an efficient data augmentation Markov Chain Monte Carlo algorithm to perform Bayesian inference. We use the posterior predictive distribution to simulate the effect of different disease control methods and their economic impact. We analyse a large outbreak of avian influenza in the Netherlands and infer the between-farm infection rate, as well as the unknown infection status of farms which were pre-emptively culled. We use our results to analyse ring-culling strategies, and conclude that although effective, ring-culling has limited impact in high-density areas.

A Bayesian nonparametric analysis for zero-inflated multivariate count data with application to microbiome study.

High-throughput sequencing technology has enabled researchers to profile microbial communities from a variety of environments, but analysis of multivariate taxon count data remains challenging. We develop a Bayesian nonparametric (BNP) regression model with zero inflation to analyse multivariate count data from microbiome studies. A BNP approach flexibly models microbial associations with covariates, such as environmental factors and clinical characteristics. The model produces estimates for probability distributions which relate microbial diversity and differential abundance to covariates, and facilitates community comparisons beyond those provided by simple statistical tests. We compare the model to simpler models and popular alternatives in simulation studies, showing, in addition to these additional community-level insights, it yields superior parameter estimates and model fit in various settings. The model's utility is demonstrated by applying it to a chronic wound microbiome data set and a Human Microbiome Project data set, where it is used to compare microbial communities present in different environments.

Oil Spill Detection in SAR Images Using Online Extended Variational Learning of Dirichlet Process Mixtures of Gamma Distributions

In this paper, we propose a Dirichlet process (DP) mixture model of Gamma distributions, which is an extension of the finite Gamma mixture model to the infinite case. In particular, we propose a novel online nonparametric Bayesian analysis method based on the infinite Gamma mixture model where the determination of the number of clusters is bypassed via an infinite number of mixture components. The proposed model is learned via an online extended variational Bayesian inference approach in a flexible way where the priors of model’s parameters are selected appropriately and the posteriors are approximated effectively in a closed form. The online setting has the advantage to allow data instances to be treated in a sequential manner, which is more attractive than batch learning especially when dealing with massive and streaming data. We demonstrated the performance and merits of the proposed statistical framework with a challenging real-world application namely oil spill detection in synthetic aperture radar (SAR) images.

A Common Atoms Model for the Bayesian Nonparametric Analysis of Nested Data

The use of large datasets for targeted therapeutic interventions requires new ways to characterize the heterogeneity observed across subgroups of a specific population. In particular, models for partially exchangeable data are needed for inference on nested datasets, where the observations are assumed to be organized in different units and some sharing of information is required to learn distinctive features of the units. In this manuscript, we propose a nested common atoms model (CAM) that is particularly suited for the analysis of nested datasets where the distributions of the units are expected to differ only over a small fraction of the observations sampled from each unit. The proposed CAM allows a two-layered clustering at the distributional and observational level and is amenable to scalable posterior inference through the use of a computationally efficient nested slice sampler algorithm. We further discuss how to extend the proposed modeling framework to handle discrete measurements, and we conduct posterior inference on a real microbiome dataset from a diet swap study to investigate how the alterations in intestinal microbiota composition are associated with different eating habits. We further investigate the performance of our model in capturing true distributional structures in the population by means of a simulation study.

BETS: The dangers of selection bias in early analyses of the coronavirus disease (COVID-19) pandemic

The coronavirus disease 2019 (COVID-19) has quickly grown from a regional outbreak in Wuhan, China, to a global pandemic. Early estimates of the epidemic growth and incubation period of COVID-19 may have been biased due to sample selection. Using detailed case reports from 14 locations in and outside mainland China, we obtained 378 Wuhan-exported cases who left Wuhan before an abrupt travel quarantine. We developed a generative model we call BETS for four key epidemiological events—Beginning of exposure, End of exposure, time of Transmission, and time of Symptom onset (BETS)—and derived explicit formulas to correct for the sample selection. We gave a detailed illustration of why some early and highly influential analyses of the COVID-19 pandemic were severely biased. All our analyses, regardless of which subsample and model were being used, point to an epidemic doubling time of two to 2.5 days during the early outbreak in Wuhan. A Bayesian nonparametric analysis further suggests that about 5% of the symptomatic cases may not develop symptoms within 14 days of infection and that men may be much more likely than women to develop symptoms within two days of infection.

Nonparametric Bayesian reliability analysis of masked data with dependent competing risks

The characteristic of dependence widely exists among different failure modes of systems, which brings extra difficulty for the reliability analysis. In this paper, a nonparametric Bayesian analysis method is proposed for dependent masked data under accelerated lifetime test with censoring. Using the copula function, the dependence structure is constructed among the competing failure modes which can be viewed as the components in a series system. By establishing the transformational relationship between the subsurvival functions and the survival functions, the estimators of components’ reliability can be derived from the nonparametric Bayesian estimators of the subsurvival functions when a Dirichlet multivariate process prior is considered. A simulation study is given to illustrate the effectiveness of the proposed methods and the influence of the degree of dependence on the performance of the estimation procedure. It is shown that the dependence between failure modes should not be ignored because ignoring the dependence may lead to serious errors. A numerical example based on a life test of rolling ball bearings is presented as an application of the proposed methodology.

The semi-Markov beta-Stacy process: a Bayesian non-parametric prior for semi-Markov processes.

The literature on Bayesian methods for the analysis of discrete-time semi-Markov processes is sparse. In this paper, we introduce the semi-Markov beta-Stacy process, a stochastic process useful for the Bayesian non-parametric analysis of semi-Markov processes. The semi-Markov beta-Stacy process is conjugate with respect to data generated by a semi-Markov process, a property which makes it easy to obtain probabilistic forecasts. Its predictive distributions are characterized by a reinforced random walk on a system of urns.

Online nonparametric Bayesian analysis of parsimonious Gaussian mixture models and scenes clustering

The mixture model is a very powerful and flexible tool in clustering analysis. Based on the Dirichlet process and parsimonious Gaussian distribution, we propose a new nonparametric mixture framework for solving challenging clustering problems. Meanwhile, the inference of the model depends on the efficient online variational Bayesian approach, which enhances the information exchange between the whole and the part to a certain extent and applies to scalable datasets. The experiments on the scene database indicate that the novel clustering framework, when combined with a convolutional neural network for feature extraction, has meaningful advantages over other models.

Nonparametric Bayesian analysis of the compound Poisson prior for support boundary recovery

Given data from a Poisson point process with intensity $(x,y)\mapston\mathbf{1}(f(x)\leq y)$, frequentist properties for the Bayesian reconstruction of the support boundary function $f$ are derived. We mainly study compound Poisson process priors with fixed intensity proving that the posterior contracts with nearly optimal rate for monotone support boundaries and adapts to Hölder smooth boundaries. We then derive a limiting shape result for a compound Poisson process prior and a function space with increasing parameter dimension. It is shown that the marginal posterior of the mean functional performs an automatic bias correction and contracts with a faster rate than the MLE. In this case, $(1-\alpha )$-credible sets are also asymptotic $(1-\alpha )$-confidence intervals. As a negative result, it is shown that the frequentist coverage of credible sets is lost for linear functions $f$ outside the function class.

Semiparametric mixed-scale models using shared Bayesian forests.

This paper demonstrates the advantages of sharing information about unknown features of covariates across multiple model components in various nonparametric regression problems including multivariate, heteroscedastic, and semicontinuous responses. In this paper, we present a methodology which allows for information to be shared nonparametrically across various model components using Bayesian sum-of-tree models. Our simulation results demonstrate that sharing of information across related model components is often very beneficial, particularly in sparse high-dimensional problems in which variable selection must be conducted. We illustrate our methodology by analyzing medical expenditure data from the Medical Expenditure Panel Survey (MEPS). To facilitate the Bayesian nonparametric regression analysis, we develop two novel models for analyzing the MEPS data using Bayesian additive regression trees-a heteroskedastic log-normal hurdle model with a "shrink-toward-homoskedasticity" priorand a gamma hurdle model.