Bayesian nonparametric (BNP) models provide elegant methods for discovering underlying latent features within a data set, but inference in such models can be slow. We exploit the fact that completely random measures, which commonly used models like the Dirichlet process and the beta-Bernoulli process can be expressed as, are decomposable into independent sub-measures. We use this decomposition to partition the latent measure into a finite measure containing only instantiated components, and an infinite measure containing all other components. We then select different inference algorithms for the two components: uncollapsed samplers mix well on the finite measure, while collapsed samplers mix well on the infinite, sparsely occupied tail. The resulting hybrid algorithm can be applied to a wide class of models, and can be easily distributed to allow scalable inference without sacrificing asymptotic convergence guarantees.
We introduce the nonparametric metadata dependent relational (NMDR) model, a Bayesian nonparametric stochastic block model for network data. The NMDR allows the entities associated with each node to have mixed membership in an unbounded collection of latent communities. Learned regression models allow these memberships to depend on, and be predicted from, arbitrary node metadata. We develop efficient MCMC algorithms for learning NMDR models from partially observed node relationships. Retrospective MCMC methods allow our sampler to work directly with the infinite stick-breaking representation o…
Indian Institute of Technology, Kanpur（印度理工学院坎普尔分校）
The latent feature relational model (LFRM) is a generative model for graph-structured data to learn a binary vector representation for each node in the graph. The binary vector denotes the node's membership in one or more communities. At its core, the LFRM miller2009nonparametric is an overlapping stochastic blockmodel, which defines the link probability between any pair of nodes as a bilinear function of their community membership vectors. Moreover, using a nonparametric Bayesian prior (Indian Buffet Process) enables learning the number of communities automatically from the data. However, des…
Deakin University（迪肯大学）；University of Michigan（密歇根大学）；Nuance Communications（Nuance通信公司）
We present a Bayesian nonparametric framework for multilevel clustering which utilizes group-level context information to simultaneously discover low-dimensional structures of the group contents and partitions groups into clusters. Using the Dirichlet process as the building block, our model constructs a product base-measure with a nested structure to accommodate content and context observations at multiple levels. The proposed model possesses properties that link the nested Dirichlet processes (nDP) and the Dirichlet process mixture models (DPM) in an interesting way: integrating out all cont…
In tracking multiple objects, it is often assumed that each observation (measurement) is originated from one and only one object. However, we may encounter a situation that each measurement may or may not be associated with multiple objects at each time step --spawning. Therefore, the association of each measurement to multiple objects is a crucial task to perform in order to track multiple objects with birth and death. In this paper, we introduce a novel Bayesian nonparametric approach that models a scenario where each observation may be drawn from an unknown number of objects for which it pr…
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, most of the commonly used Markov chain Monte Carlo (MCMC) methods are not suitable. 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…
Pohang University of Science and Technology（浦项科技大学）
Normalized random measures (NRMs) provide a broad class of discrete random measures that are often used as priors for Bayesian nonparametric models. Dirichlet process is a well-known example of NRMs. Most of posterior inference methods for NRM mixture models rely on MCMC methods since they are easy to implement and their convergence is well studied. However, MCMC often suffers from slow convergence when the acceptance rate is low. Tree-based inference is an alternative deterministic posterior inference method, where Bayesian hierarchical clustering (BHC) or incremental Bayesian hierarchical cl…
Institute of Software, Chinese Academy of Sciences（中国科学院软件研究所）；Huawei Noah’s Ark Lab（华为诺亚方舟实验室）
In real world machine learning applications, testing data may contain some meaningful new categories that have not been seen in labeled training data. To simultaneously recognize new data categories and assign most appropriate category labels to the data actually from known categories, existing models assume the number of unknown new categories is pre-specified, though it is difficult to determine in advance. In this paper, we propose a Bayesian nonparametric topic model to automatically infer this number, based on the hierarchical Dirichlet process and the notion of latent Dirichlet allocatio…
The goal of phylodynamics, an area on the intersection of phylogenetics and population genetics, is to reconstruct population size dynamics from genetic data. Recently, a series of nonparametric Bayesian methods have been proposed for such demographic reconstructions. These methods rely on prior specifications based on Gaussian processes and proceed by approximating the posterior distribution of population size trajectories via Markov chain Monte Carlo (MCMC) methods. In this paper, we adapt an integrated nested Laplace approximation (INLA), a recently proposed approximate Bayesian inference f…
Latent class analysis is used to perform model based clustering for multivariate categorical responses. Selection of the variables most relevant for clustering is an important task which can affect the quality of clustering considerably. This work considers a Bayesian approach for selecting the number of clusters and the best clustering variables. The main idea is to reformulate the problem of group and variable selection as a probabilistically driven search over a large discrete space using Markov chain Monte Carlo (MCMC) methods. This approach results in estimates of degree of relevance of e…
Group factor analysis (GFA) methods have been widely used to infer the common structure and the group-specific signals from multiple related datasets in various fields including systems biology and neuroimaging. To date, most available GFA models require Gibbs sampling or slice sampling to perform inference, which prevents the practical application of GFA to large-scale data. In this paper we present an efficient collapsed variational inference (CVI) algorithm for the nonparametric Bayesian group factor analysis (NGFA) model built upon an hierarchical beta Bernoulli process. Our CVI algorithm …
We investigate the class of $σ$-stable Poisson-Kingman random probability measures (RPMs) in the context of Bayesian nonparametric mixture modeling. This is a large class of discrete RPMs which encompasses most of the the popular discrete RPMs used in Bayesian nonparametrics, such as the Dirichlet process, Pitman-Yor process, the normalized inverse Gaussian process and the normalized generalized Gamma process. We show how certain sampling properties and marginal characterizations of $σ$-stable Poisson-Kingman RPMs can be usefully exploited for devising a Markov chain Monte Carlo (MCMC) algorit…
Semi-supervised clustering is the task of clustering data points into clusters where only a fraction of the points are labelled. The true number of clusters in the data is often unknown and most models require this parameter as an input. Dirichlet process mixture models are appealing as they can infer the number of clusters from the data. However, these models do not deal with high dimensional data well and can encounter difficulties in inference. We present a novel nonparameteric Bayesian kernel based method to cluster data points without the need to prespecify the number of clusters or to mo…
We consider priors for several nonparametric Bayesian models which use finite random series with a random number of terms. The prior is constructed through distributions on the number of basis functions and the associated coefficients. We derive a general result on the construction of an appropriate sieve and obtain adaptive posterior contraction rates for all smoothness levels of the function in the true model. We apply this general result on several statistical problems such as signal processing, density estimation, nonparametric additive regression, classification, spectral density estimati…
Given discrete time observations over a growing time interval, we consider a nonparametric Bayesian approach to estimation of the Lévy density of a Lévy process belonging to a flexible class of infinite activity subordinators. Posterior inference is performed via MCMC, and we circumvent the problem of the intractable likelihood via the data augmentation device, that in our case relies on bridge process sampling via Gamma process bridges. Our approach also requires the use of a new infinite-dimensional form of a reversible jump MCMC algorithm. We show that our method leads to good practical res…
In this paper we propose the first non-parametric Bayesian model using Gaussian Processes to make inference on Poisson Point Processes without resorting to gridding the domain or to introducing latent thinning points. Unlike competing models that scale cubically and have a squared memory requirement in the number of data points, our model has a linear complexity and memory requirement. We propose an MCMC sampler and show that our model is faster, more accurate and generates less correlated samples than competing models on both synthetic and real-life data. Finally, we show that our model easil…
We consider the Bayesian mixture of finite mixtures (MFMs) and Dirichlet process mixture (DPM) models for clustering. Recent asymptotic theory has established that DPMs overestimate the number of clusters for large samples and that estimators from both classes of models are inconsistent for the number of clusters under misspecification, but the implications for finite sample analyses are unclear. The final reported estimate after fitting these models is often a single representative clustering obtained using an MCMC summarisation technique, but it is unknown how well such a summary estimates t…
Spatiotemporal Clustering with Neyman-Scott Processes via Connections to Bayesian Nonparametric Mixture Models
University of Michigan（密歇根大学）；Stanford University（斯坦福大学）；New York University（纽约大学）；Flatiron Institute（弗拉蒂伦研究所）；Wu Tsai Neurosciences Institute（吴蔡神经科学研究所）
Most of applied statistics involves regression analysis of data. This paper presents a stand-alone and menu-driven software package, Bayesian Regression: Nonparametric and Parametric Models. Currently, this package gives the user a choice from 83 Bayesian models for data analysis. They include 47 Bayesian nonparametric (BNP) infinite-mixture regression models; 5 BNP infinite-mixture models for density estimation; and 31 normal random effects models (HLMs), including normal linear models. Each of the 78 regression models handles either a continuous, binary, or ordinal dependent variable, and ca…
We propose a Bayesian nonparametric approach for the noise reduction of a given chaotic time series contaminated by dynamical noise, based on Markov Chain Monte Carlo methods (MCMC). The underlying unknown noise process (possibly) exhibits heavy tailed behavior. We introduce the Dynamic Noise Reduction Replicator (DNRR) model with which we reconstruct the unknown dynamic equations and in parallel we replicate the dynamics under reduced noise level dynamical perturbations. The dynamic noise reduction procedure is demonstrated specifically in the case of polynomial maps. Simulations based on syn…
Topic models, and more specifically the class of Latent Dirichlet Allocation (LDA), are widely used for probabilistic modeling of text. MCMC sampling from the posterior distribution is typically performed using a collapsed Gibbs sampler. We propose a parallel sparse partially collapsed Gibbs sampler and compare its speed and efficiency to state-of-the-art samplers for topic models on five well-known text corpora of differing sizes and properties. In particular, we propose and compare two different strategies for sampling the parameter block with latent topic indicators. The experiments show th…
We propose a Bayesian nonparametric model based on Markov Chain Monte Carlo (MCMC) methods for unveiling the structure of the invariant global stable manifold from observed time-series data. The underlying unknown dynamical process is possibly contaminated by additive noise. We introduce the Stable Manifold Geometric Stick Breaking Reconstruction (SM-GSBR) model with which we reconstruct the unknown dynamic equations and in parallel we estimate the global structure of the perturbed stable manifold. Our method works for noninvertible maps without modifications. The stable manifold estimation pr…
In this paper we propose a new adaptive wavelet denoising methodology using complex wavelets. The method is based on a fully Bayesian hierarchical model in the complex wavelet domain that uses a bivariate mixture prior on the wavelet coefficients. The heart of the procedure is computational, where the posterior mean is computed through Markov chain Monte Carlo (MCMC) simulations. We show that the method has good performance, as demonstrated by simulations on the well-known test functions and by comparison to a well-established complex wavelet-based denoising procedure. An application to real-l…
We present a Bayesian nonparametric system reliability model which scales well and provides a great deal of flexibility in modeling. The Bayesian approach naturally handles the disparate amounts of component and subsystem data that may exist. However, traditional Bayesian reliability models are quite computationally complex, relying on MCMC techniques. Our approach utilizes the conjugate properties of the beta-Stacy process, which is the fundamental building block of our model. These individual models are linked together using a method of moments estimation approach. This model is computationa…
Intensity estimation is a common problem in statistical analysis of spatial point pattern data. This paper proposes a nonparametric Bayesian method for estimating the spatial point process intensity based on mixture of finite mixture (MFM) model. MFM approach leads to a consistent estimate of the intensity of spatial point patterns in different areas while considering heterogeneity. An efficient Markov chain Monte Carlo (MCMC) algorithm is proposed for our method. Extensive simulation studies are carried out to examine empirical performance of the proposed method. The usage of our proposed met…
This paper concerns the introduction of a new Markov Chain Monte Carlo scheme for posterior sampling in Bayesian nonparametric mixture models with priors that belong to the general Poisson-Kingman class. We present a novel compact way of representing the infinite dimensional component of the model such that while explicitly representing this infinite component it has less memory and storage requirements than previous MCMC schemes. We describe comparative simulation results demonstrating the efficacy of the proposed MCMC algorithm against existing marginal and conditional MCMC samplers.
We present the Gaussian process density sampler (GPDS), an exchangeable generative model for use in nonparametric Bayesian density estimation. Samples drawn from the GPDS are consistent with exact, independent samples from a distribution defined by a density that is a transformation of a function drawn from a Gaussian process prior. Our formulation allows us to infer an unknown density from data using Markov chain Monte Carlo, which gives samples from the posterior distribution over density functions and from the predictive distribution on data space. We describe two such MCMC methods. Both me…
Genetic sequence data are well described by hidden Markov models (HMMs) in which latent states correspond to clusters of similar mutation patterns. Theory from statistical genetics suggests that these HMMs are nonhomogeneous (their transition probabilities vary along the chromosome) and have large support for self transitions. We develop a new nonparametric model of genetic sequence data, based on the hierarchical Dirichlet process, which supports these self transitions and nonhomogeneity. Our model provides a parameterization of the genetic process that is more parsimonious than other more ge…
We propose a Bayesian nonparametric mixture model for the reconstruction and prediction from observed time series data, of discretized stochastic dynamical systems, based on Markov Chain Monte Carlo methods (MCMC). Our results can be used by researchers in physical modeling interested in a fast and accurate estimation of low dimensional stochastic models when the size of the observed time series is small and the noise process (perhaps) is non-Gaussian. The inference procedure is demonstrated specifically in the case of polynomial maps of arbitrary degree and when a Geometric Stick Breaking mix…
In many applications, such as economics, operations research and reinforcement learning, one often needs to estimate a multivariate regression function f subject to a convexity constraint. For example, in sequential decision processes the value of a state under optimal subsequent decisions may be known to be convex or concave. We propose a new Bayesian nonparametric multivariate approach based on characterizing the unknown regression function as the max of a random collection of unknown hyperplanes. This specification induces a prior with large support in a Kullback-Leibler sense on the space …
In this paper, we propose a nonparametric Bayesian approach for Lindsey and penalized Gaussian mixtures methods. We compare these methods with the Dirichlet process mixture model. Our approach is a Bayesian nonparametric method not based solely on a parametric family of probability distributions. Thus, the fitted models are more robust to model misspecification. Also, with the Bayesian approach, we have the entire posterior distribution of our parameter of interest; it can be summarized through credible intervals, mean, median, standard deviation, quantiles, etc. The Lindsey, penalized Gaussia…
Recently, the Bayesian nonparametric approach in survival studies attracts much more attentions. Because of multi modality in survival data, the mixture models are very common in this field. One of the famous priors on Bayesian nonparametric models is Dirichlet process prior. In this paper we introduce a Bayesian nonparametric mixture model with Burr distribution(Burr type XII) as the kernel of mixture model. Since the Burr distribution shares good properties of common distributions on survival analysis, it has more flexibility than other distributions. By applying this model to simulated and …
A common approach to analyze a covariate-sample count matrix, an element of which represents how many times a covariate appears in a sample, is to factorize it under the Poisson likelihood. We show its limitation in capturing the tendency for a covariate present in a sample to both repeat itself and excite related ones. To address this limitation, we construct negative binomial factor analysis (NBFA) to factorize the matrix under the negative binomial likelihood, and relate it to a Dirichlet-multinomial distribution based mixed-membership model. To support countably infinite factors, we propos…
Criteo AI Lab（Criteo人工智能实验室）；IRIT（图卢兹计算机科学研究所）；Université de Toulouse（图卢兹大学）；CNRS（法国国家科学研究中心）
Binary data matrices can represent many types of data such as social networks, votes, or gene expression. In some cases, the analysis of binary matrices can be tackled with nonnegative matrix factorization (NMF), where the observed data matrix is approximated by the product of two smaller nonnegative matrices. In this context, probabilistic NMF assumes a generative model where the data is usually Bernoulli-distributed. Often, a link function is used to map the factorization to the $[0,1]$ range, ensuring a valid Bernoulli mean parameter. However, link functions have the potential disadvantage …
University of Massachusetts Amherst（马萨诸塞大学阿默斯特分校）；The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Microsoft Research New York（微软研究院纽约）
We introduce a new dynamical system for sequentially observed multivariate count data. This model is based on the gamma--Poisson construction---a natural choice for count data---and relies on a novel Bayesian nonparametric prior that ties and shrinks the model parameters, thus avoiding overfitting. We present an efficient MCMC inference algorithm that advances recent work on augmentation schemes for inference in negative binomial models. Finally, we demonstrate the model's inductive bias using a variety of real-world data sets, showing that it exhibits superior predictive performance over othe…
In this article a flexible Bayesian non-parametric model is proposed for non-homogeneous hidden Markov models. The model is developed through the amalgamation of the ideas of hidden Markov models and predictor dependent stick-breaking processes. Computation is carried out using auxiliary variable representation of the model which enable us to perform exact MCMC sampling from the posterior. Furthermore, the model is extended to the situation when the predictors can simultaneously in influence the transition dynamics of the hidden states as well as the emission distribution. Estimates of few ste…
Although multivariate count data are routinely collected in many application areas, there is surprisingly little work developing flexible models for characterizing their dependence structure. This is particularly true when interest focuses on inferring the conditional independence graph. In this article, we propose a new class of pairwise Markov random field-type models for the joint distribution of a multivariate count vector. By employing a novel type of transformation, we avoid restricting to non-negative dependence structures or inducing other restrictions through truncations. Taking a Bay…
A fully Bayesian approach is proposed for ultrahigh-dimensional nonparametric additive models in which the number of additive components may be larger than the sample size, though ideally the true model is believed to include only a small number of components. Bayesian approaches can conduct stochastic model search and fulfill flexible parameter estimation by stochastic draws. The theory shows that the proposed model selection method has satisfactory properties. For instance, when the hyperparameter associated with the model prior is correctly specified, the true model has posterior probabilit…
This paper concerns the use of Markov chain Monte Carlo methods for posterior sampling in Bayesian nonparametric mixture models with normalized random measure priors. Making use of some recent posterior characterizations for the class of normalized random measures, we propose novel Markov chain Monte Carlo methods of both marginal type and conditional type. The proposed marginal samplers are generalizations of Neal's well-regarded Algorithm 8 for Dirichlet process mixture models, whereas the conditional sampler is a variation of those recently introduced in the literature. For both the margina…
Pólya trees fix partitions and use random probabilities in order to construct random probability measures. With quantile pyramids we instead fix probabilities and use random partitions. For nonparametric Bayesian inference we use a prior which supports piecewise linear quantile functions, based on the need to work with a finite set of partitions, yet we show that the limiting version of the prior exists. We also discuss and investigate an alternative model based on the so-called substitute likelihood. Both approaches factorize in a convenient way leading to relatively straightforward analysis …
This article is motivated by the problem of studying the joint effect of different chemical exposures on human health outcomes. This is essentially a nonparametric regression problem, with interest being focused not on a black box for prediction but instead on selection of main effects and interactions. For interpretability, we decompose the expected health outcome into a linear main effect, pairwise interactions, and a non-linear deviation. Our interest is in model selection for these different components, accounting for uncertainty and addressing non-identifability between the linear and non…
We introduce state-space models where the functionals of the observational and the evolutionary equations are unknown, and treated as random functions evolving with time. Thus, our model is nonparametric and generalizes the traditional parametric state-space models. This random function approach also frees us from the restrictive assumption that the functional forms, although time-dependent, are of fixed forms. The traditional approach of assuming known, parametric functional forms is questionable, particularly in state-space models, since the validation of the assumptions require data on both…
Basketball shot location data provide valuable summary information regarding players to coaches, sports analysts, fans, statisticians, as well as players themselves. Represented by spatial points, such data are naturally analyzed with spatial point process models. We present a novel nonparametric Bayesian method for learning the underlying intensity surface built upon a combination of Dirichlet process and Markov random field. Our method has the advantage of effectively encouraging local spatial homogeneity when estimating a globally heterogeneous intensity surface. Posterior inferences are pe…
Cost-effectiveness analyses (CEAs) are at the center of health economic decision making. While these analyses help policy analysts and economists determine coverage, inform policy, and guide resource allocation, they are statistically challenging for several reasons. Cost and effectiveness are correlated and follow complex joint distributions which are difficult to capture parametrically. Effectiveness (often measured as increased survival time) and accumulated cost tends to be right-censored in many applications. Moreover, CEAs are often conducted using observational data with non-random trea…
Bayesian nonparametric regression with dependent wavelets has dual shrinkage properties: there is shrinkage through a dependent prior put on functional differences, and shrinkage through the setting of most of the wavelet coefficients to zero through Bayesian variable selection methods. The methodology can deal with unequally spaced data and is efficient because of the existence of fast moves in model space for the MCMC computation. The methodology is illustrated on the problem of modeling the oscillations of Cepheid variable stars; these are a class of pulsating variable stars with the useful…
Even though Nearest Neighbor Gaussian Processes (NNGP) alleviate considerably MCMC implementation of Bayesian space-time models, they do not solve the convergence problems caused by high model dimension. Frugal alternatives such as response or collapsed algorithms are an answer.gree Our approach is to keep full data augmentation but to try and make it more efficient. We present two strategies to do so. The first scheme is to pay a particular attention to the seemingly trivial fixed effects of the model. We show empirically that re-centering the latent field on the intercept critically improves…
This paper introduces a flexible Bayesian nonparametric Item Response Theory (IRT) model, which applies to dichotomous or polytomous item responses, and which can apply to either unidimensional or multidimensional scaling. This is an infinite-mixture IRT model, with person ability and item difficulty parameters, and with a random intercept parameter that is assigned a mixing distribution, with mixing weights a probit function of other person and item parameters. As a result of its flexibility, the Bayesian nonparametric IRT model can provide outlier-robust estimation of the person ability para…
This article introduces a flexible and adaptive nonparametric method for estimating the association between multiple covariates and power spectra of multiple time series. The proposed approach uses a Bayesian sum of trees model to capture complex dependencies and interactions between covariates and the power spectrum, which are often observed in studies of biomedical time series. Local power spectra corresponding to terminal nodes within trees are estimated nonparametrically using Bayesian penalized linear splines. The trees are considered to be random and fit using a Bayesian backfitting Mark…
Deep kernel learning is a promising combination of deep neural networks and nonparametric function learning. However, as a data driven approach, the performance of deep kernel learning can still be restricted by scarce or insufficient data, especially in extrapolation tasks. To address these limitations, we propose Physics Informed Deep Kernel Learning (PI-DKL) that exploits physics knowledge represented by differential equations with latent sources. Specifically, we use the posterior function sample of the Gaussian process as the surrogate for the solution of the differential equation, and co…
We propose autoregressive Bayesian semi-parametric models for waiting times between recurrent events. The aim is two-fold: inference on the effect of possibly time-varying covariates on the gap times and clustering of individuals based on the time trajectory of the recurrent event. Time-dependency between gap times is taken into account through the specification of an autoregressive component for the random effects parameters influencing the response at different times. The order of the autoregression may be assumed unknown and object of inference and we consider two alternative approaches to …
Non-negative Matrix Factorization (NMF) is a popular tool for data exploration. Bayesian NMF promises to also characterize uncertainty in the factorization. Unfortunately, current inference approaches such as MCMC mix slowly and tend to get stuck on single modes. We introduce a novel approach using rapidly-exploring random trees (RRTs) to asymptotically cover regions of high posterior density. These are placed in a principled Bayesian framework via an online extension to nonparametric variational inference. On experiments on real and synthetic data, we obtain greater coverage of the posterior …
State space models are well-known for their versatility in modeling dynamic systems that arise in various scientific disciplines. Although parametric state space models are well studied, nonparametric approaches are much less explored in comparison. In this article we propose a novel Bayesian nonparametric approach to state space modeling assuming that both the observational and evolutionary functions are unknown and are varying with time; crucially, we assume that the unknown evolutionary equation describes dynamic evolution of some latent circular random variable. Based on appropriate kernel…
