Using Kalman techniques, it is possible to perform optimal estimation in linear Gaussian state-space models. We address here the case where the noise probability density functions are of unknown functional form. A flexible Bayesian nonparametric noise model based on Dirichlet process mixtures is introduced. Efficient Markov chain Monte Carlo and Sequential Monte Carlo methods are then developed to perform optimal batch and sequential estimation in such contexts. The algorithms are applied to blind deconvolution and change point detection. Experimental results on synthetic and real data demonstrate the efficiency of this approach in various contexts.
Dirichlet Process Mixtures (DPMs) are a popular class of statistical models to perform density estimation and clustering. However, when the data available have a distribution evolving over time, such models are inadequate. We introduce here a class of time-varying DPMs which ensures that at each time step the random distribution follows a DPM model. Our model relies on an intuitive and simple generalized Polya urn scheme. Inference is performed using Markov chain Monte Carlo and Sequential Monte Carlo. We demonstrate our model on various applications.
Massachusetts Institute of Technology（麻省理工学院）；Harvard University（哈佛大学）
The Dirichlet process (DP) is a fundamental mathematical tool for Bayesian nonparametric modeling, and is widely used in tasks such as density estimation, natural language processing, and time series modeling. Although MCMC inference methods for the DP often provide a gold standard in terms asymptotic accuracy, they can be computationally expensive and are not obviously parallelizable. We propose a reparameterization of the Dirichlet process that induces conditional independencies between the atoms that form the random measure. This conditional independence enables many of the Markov chain tra…
Indian Institute Of Science（印度科学学院）；IBM Research Labs（IBM研究实验室）
Dirichlet Process(DP) is a Bayesian non-parametric prior for infinite mixture modeling, where the number of mixture components grows with the number of data items. The Hierarchical Dirichlet Process (HDP), is an extension of DP for grouped data, often used for non-parametric topic modeling, where each group is a mixture over shared mixture densities. The Nested Dirichlet Process (nDP), on the other hand, is an extension of the DP for learning group level distributions from data, simultaneously clustering the groups. It allows group level distributions to be shared across groups in a non-parame…
Bayesian mixture models are widely applied for unsupervised learning and exploratory data analysis. Markov chain Monte Carlo based on Gibbs sampling and split-merge moves are widely used for inference in these models. However, both methods are restricted to limited types of transitions and suffer from torpid mixing and low accept rates even for problems of modest size. We propose a method that considers a broader range of transitions that are close to equilibrium by exploiting multiple chains in parallel and using the past states adaptively to inform the proposal distribution. The method signi…
We present a hybrid algorithm for Bayesian topic models that combines the efficiency of sparse Gibbs sampling with the scalability of online stochastic inference. We used our algorithm to analyze a corpus of 1.2 million books (33 billion words) with thousands of topics. Our approach reduces the bias of variational inference and generalizes to many Bayesian hidden-variable models.
We study nonparametric Bayesian inference with location mixtures of the Laplace density and a Dirichlet process prior on the mixing distribution. We derive a contraction rate of the corresponding posterior distribution, both for the mixing distribution relative to the Wasserstein metric and for the mixed density relative to the Hellinger and $L_q$ metrics.
We show that rate-adaptive multivariate density estimation can be performed using Bayesian methods based on Dirichlet mixtures of normal kernels with a prior distribution on the kernel's covariance matrix parameter. We derive sufficient conditions on the prior specification that guarantee convergence to a true density at a rate that is optimal minimax for the smoothness class to which the true density belongs. No prior knowledge of smoothness is assumed. The sufficient conditions are shown to hold for the Dirichlet location mixture of normals prior with a Gaussian base measure and an inverse-W…
It is shown that a simple Dirichlet process mixture of multivariate normals offers Bayesian density estimation with adaptive posterior convergence rates. Toward this, a novel sieve for non-parametric mixture densities is explored, and its rate adaptability to various smoothness classes of densities in arbitrary dimension is demonstrated. This sieve construction is expected to offer a substantial technical advancement in studying Bayesian non-parametric mixture models based on stick-breaking priors.
The Bayesian nonparametric inference and Dirichlet process are popular tools in statistical methodologies. In this paper, we employ the Dirichlet process in hypothesis testing to propose a Bayesian nonparametric chi-squared goodness-of-fit test. In our Bayesian nonparametric approach, we consider the Dirichlet process as the prior for the distribution of data and carry out the test based on the Kullback-Leibler distance between the updated Dirichlet process and the hypothesized distribution F0. We prove that this distance asymptotically converges to the same chi-squared distribution as the chi…
For data assumed to come from a finite mixture with an unknown number of components, it has become common to use Dirichlet process mixtures (DPMs) not only for density estimation, but also for inferences about the number of components. The typical approach is to use the posterior distribution on the number of components occurring so far --- that is, the posterior on the number of clusters in the observed data. However, it turns out that this posterior is not consistent --- it does not converge to the true number of components. In this note, we give an elementary demonstration of this inconsist…
The influence of Dirichlet process mixture is ubiquitous in the Bayesian nonparametrics literature. But sampling from its posterior distribution remains a challenge, despite the advent of various Markov chain Monte Carlo methods. The primary challenge is the infinite-dimensional setup, and even if the infinite-dimensional random measure is integrated out, high-dimensionality and discreteness still remain difficult issues to deal with. In this article, exploiting the key ideas proposed in Bhattacharya (2021b), we propose a novel methodology for drawing iid realizations from posteriors of Dirich…
In this paper, we describe a Bayesian nonparametric approach to make inference for a bivariate spherically symmetric distribution. We consider a Dirichlet invariant process prior on the set of all bivariate spherically symmetric distributions and we derive the Dirichlet invariant process posterior. Indeed, our approach is an extension of Dirichlet invariant process for the symmetric distributions on the real line to a bivariate spherically symmetric distribution where the underlying distribution is invariant under a finite group of rotations. Moreover, we obtain the Dirichlet invariant process…
Dirichlet process mixtures are flexible non-parametric models, particularly suited to density estimation and probabilistic clustering. In this work we study the posterior distribution induced by Dirichlet process mixtures as the sample size increases, and more specifically focus on consistency for the unknown number of clusters when the observed data are generated from a finite mixture. Crucially, we consider the situation where a prior is placed on the concentration parameter of the underlying Dirichlet process. Previous findings in the literature suggest that Dirichlet process mixtures are t…
In his 1986 book, Aitchison explains that compositional data is regularly mishandled in statistical analyses, a pattern that continues to this day. The Dirichlet Type I distribution is a multivariate distribution commonly used to model a set of proportions that sum to one. Aitchinson goes on to lament the difficulties of Dirichlet modelling and the scarcity of alternatives. While he addresses the second of these issues, we address the first. The Dirichlet Type II distribution is a transformation of the Dirichlet Type I distribution and is a multivariate distribution on the positive real number…
We present an alternative approach to the Bayesian nonparametric analysis of conditional species richness under two-parameter Poisson Dirichlet priors. We rely on a known characterization by deletion of classes property and on results for Beta-Binomial distributions. Besides leading to simplified and much more direct proofs, our proposal provides a new scale mixture representation of the conditional asymptotic law.
For extreme value estimation we propose to use a model with a Dirichlet process mixture of gamma densities in the center and generalized Pareto densities for the tails. Due to the randomness in the center and a heavy tailed density in the tails density estimation and posterior inference for high quantiles are possible. The approach can be used in a "default" manner on the positive reals because it works when prior information is unavailable. The proposed model can be easy to implement and a sensitivity analysis is provided. We applied the proposed model for simulated and real data sets.
We study the rates of convergence of the posterior distribution for Bayesian density estimation with Dirichlet mixtures of normal distributions as the prior. The true density is assumed to be twice continuously differentiable. The bandwidth is given a sequence of priors which is obtained by scaling a single prior by an appropriate order. In order to handle this problem, we derive a new general rate theorem by considering a countable covering of the parameter space whose prior probabilities satisfy a summability condition together with certain individual bounds on the Hellinger metric entropy. …
Mixture models are a natural choice in many applications, but it can be difficult to place an a priori upper bound on the number of components. To circumvent this, investigators are turning increasingly to Dirichlet process mixture models (DPMMs). It is therefore important to develop an understanding of the strengths and weaknesses of this approach. This work considers the MAP (maximum a posteriori) clustering for the Gaussian DPMM (where the cluster means have Gaussian distribution and, for each cluster, the observations within the cluster have Gaussian distribution). Some desirable propertie…
Asymptotic theory of tail index estimation has been studied extensively in the frequentist literature on extreme values, but rarely in the Bayesian context. We investigate whether popular Bayesian kernel mixture models are able to support heavy tailed distributions and consistently estimate the tail index. We show that posterior inconsistency in tail index is surprisingly common for both parametric and nonparametric mixture models. We then present a set of sufficient conditions under which posterior consistency in tail index can be achieved, and verify these conditions for Pareto mixture model…
In recent years, multiple hypothesis testing has come to the forefront of statistical research, ostensibly in relation to applications in genomics and some other emerging fields. The false discovery rate (FDR) and its variants provide very important notions of errors in this context comparable to the role of error probabilities in classical testing problems. Accurate estimation of positive FDR (pFDR), a variant of the FDR, is essential in assessing and controlling this measure. In a recent paper, the authors proposed a model-based nonparametric Bayesian method of estimation of the pFDR functio…
We introduce a new nonlinear model for classification, in which we model the joint distribution of response variable, y, and covariates, x, non-parametrically using Dirichlet process mixtures. We keep the relationship between y and x linear within each component of the mixture. The overall relationship becomes nonlinear if the mixture contains more than one component. We use simulated data to compare the performance of this new approach to a simple multinomial logit (MNL) model, an MNL model with quadratic terms, and a decision tree model. We also evaluate our approach on a protein fold classi…
Large sample size equivalence between the celebrated {\it approximated} Good-Turing estimator of the probability to discover a species already observed a certain number of times (Good, 1953) and the modern Bayesian nonparametric counterpart has been recently established by virtue of a particular smoothing rule based on the two-parameter Poisson-Dirichlet model. Here we improve on this result showing that, for any finite sample size, when the population frequencies are assumed to be selected from a superpopulation with two-parameter Poisson-Dirichlet distribution, then Bayesian nonparametric es…
We investigate the geometry of the maximal a posteriori (MAP) partition in the Bayesian Mixture Model where the component and the base distributions are chosen from conjugate exponential families. We prove that in this case the clusters are separated by the contour lines of a linear functional of the sufficient statistic. As a particular example, we describe Bayesian Mixture of Normals with Normal-inverse-Wishart prior on the component mean and covariance, in which the clusters in any MAP partition are separated by a quadratic surface. In connection with results of Rajkowski (2018), where the …
For two vast families of mixture distributions and a given prior, we provide unified representations of posterior and predictive distributions. Model applications presented include bivariate mixtures of Gamma distributions labelled as Kibble-type, non-central Chi-square and F distributions, the distribution of $R^2$ in multiple regression, variance mixture of normal distributions, and mixtures of location-scale exponential distributions including the multivariate Lomax distribution. An emphasis is also placed on analytical representations and the relationships with a host of existing distribut…
A Poisson mixture is one of the practically important models in computer science, biology, and sociology. However, the theoretical property has not been studied because the posterior distribution can not be approximated by any normal distribution. Such a model is called singular and it is known that Real Log Canonical Threshold (RLCT) is equal to the coefficient of the asymptotically main term of the Bayesian generalization error. In this paper, we derive RLCT of a simplex Vandermonde matrix type singularity which is equal to that of a Poisson mixture in general cases.
We introduce a new class of nonparametric prior distributions on the space of continuously varying densities, induced by Dirichlet process mixtures which diffuse in time. These select time-indexed random functions without jumps, whose sections are continuous or discrete distributions depending on the choice of kernel. The construction exploits the widely used stick-breaking representation of the Dirichlet process and induces the time dependence by replacing the stick-breaking components with one-dimensional Wright-Fisher diffusions. These features combine appealing properties of the model, inh…
The efficiency of two Bayesian order estimators is studied. By using nonparametric techniques, we prove new underestimation and overestimation bounds. The results apply to various models, including mixture models. In this case, the errors are shown to be $O(e^{-an})$ and $O((\log n)^b/\sqrt{n})$ ($a,b>0$), respectively.
In this paper we provide general conditions to check on the model and the prior to derive posterior concentration rates for data-dependent priors (or empirical Bayes approaches). We aim at providing conditions that are close to the conditions provided in the seminal paper by Ghosal and van der Vaart (2007a). We then apply the general theorem to two different settings: the estimation of a density using Dirichlet process mixtures of Gaussian random variables with base measure depending on some empirical quantities and the estimation of the intensity of a counting process under the Aalen model. A…
NTT DATA Mathematical Systems Inc.（NTT数据数学系统公司）；Tokyo Institute of Technology（东京工业大学）
Latent Dirichlet allocation (LDA) obtains essential information from data by using Bayesian inference. It is applied to knowledge discovery via dimension reducing and clustering in many fields. However, its generalization error had not been yet clarified since it is a singular statistical model where there is no one-to-one mapping from parameters to probability distributions. In this paper, we give the exact asymptotic form of its generalization error and marginal likelihood, by theoretical analysis of its learning coefficient using algebraic geometry. The theoretical result shows that the Bay…
Dirichlet process mixture models (DPMM) play a central role in Bayesian nonparametrics, with applications throughout statistics and machine learning. DPMMs are generally used in clustering problems where the number of clusters is not known in advance, and the posterior distribution is treated as providing inference for this number. Recently, however, it has been shown that the DPMM is inconsistent in inferring the true number of components in certain cases. This is an asymptotic result, and it would be desirable to understand whether it holds with finite samples, and to more fully understand t…
Multinomial mixtures are widely used in the information engineering field, however, their mathematical properties are not yet clarified because they are singular learning models. In fact, the models are non-identifiable and their Fisher information matrices are not positive definite. In recent years, the mathematical foundation of singular statistical models are clarified by using algebraic geometric methods. In this paper, we clarify the real log canonical thresholds and multiplicities of the multinomial mixtures and elucidate their asymptotic behaviors of generalization error and free energy…
NICTA（澳大利亚国家信息与通信技术研究院）；Australian National University（澳大利亚国立大学）
The two parameter Poisson-Dirichlet Process (PDP), a generalisation of the Dirichlet Process, is increasingly being used for probabilistic modelling in discrete areas such as language technology, bioinformatics, and image analysis. There is a rich literature about the PDP and its derivative distributions such as the Chinese Restaurant Process (CRP). This article reviews some of the basic theory and then the major results needed for Bayesian modelling of discrete problems including details of priors, posteriors and computation. The PDP allows one to build distributions over countable partitions…
A classic approach for learning Bayesian networks from data is to identify a maximum a posteriori (MAP) network structure. In the case of discrete Bayesian networks, MAP networks are selected by maximising one of several possible Bayesian Dirichlet (BD) scores; the most famous is the Bayesian Dirichlet equivalent uniform (BDeu) score from Heckerman et al (1995). The key properties of BDeu arise from its uniform prior over the parameters of each local distribution in the network, which makes structure learning computationally efficient; it does not require the elicitation of prior knowledge fro…
Mixture models are one of the most widely used statistical tools when dealing with data from heterogeneous populations. This paper considers the long-standing debate over finite mixture and infinite mixtures and brings the two modelling strategies together, by showing that a finite mixture is simply a realization of a point process. Following a Bayesian nonparametric perspective, we introduce a new class of prior: the Normalized Independent Point Processes. We investigate the probabilistic properties of this new class. Moreover, we design a conditional algorithm for finite mixture models with …
