Finite mixture models are an important tool in the statistical analysis of data, for example in data clustering. The optimal parameters of a mixture model are usually computed by maximizing the log-likelihood functional via the Expectation-Maximization algorithm. We propose an alternative approach based on the theory of Mean Field Games, a class of differential games with an infinite number of agents. We show that the solution of a finite state space multi-population Mean Field Games system characterizes the critical points of the log-likelihood functional for a Bernoulli mixture. The approach is then generalized to mixture models of categorical distributions. Hence, the Mean Field Games approach provides a method to compute the parameters of the mixture model, and we show its application to some standard examples in cluster analysis.
A Bernoulli Mixture Model (BMM) is a finite mixture of random binary vectors with independent dimensions. The problem of clustering BMM data arises in a variety of real-world applications, ranging from population genetics to activity analysis in social networks. In this paper, we analyze the clusterability of BMMs from a theoretical perspective, when the number of clusters is unknown. In particular, we stipulate a set of conditions on the sample complexity and dimension of the model in order to guarantee the Probably Approximately Correct (PAC)-clusterability of a dataset. To the best of our k…
CSIRO Mathematical and Information Sciences（澳大利亚联邦科学与工业研究组织数学与信息科学研究所）
A novel method for estimating Bayesian network (BN) parameters from data is presented which provides improved performance on test data. Previous research has shown the value of representing conditional probability distributions (CPDs) via neural networks(Neal 1992), noisy-OR gates (Neal 1992, Diez 1993)and decision trees (Friedman and Goldszmidt 1996).The Bernoulli mixture network (BMN) explicitly represents the CPDs of discrete BN nodes as mixtures of local distributions,each having a different set of parents.This increases the space of possible structures which can be considered,enabling the…
We present a new method of generating mixture models for data with categorical attributes. The keys to this approach are an entropy-based density metric in categorical space and annealing of high-entropy/low-density components from an initial state with many components. Pruning of low-density components using the entropy-based density allows GALILEO to consistently find high-quality clusters and the same optimal number of clusters. GALILEO has shown promising results on a range of test datasets commonly used for categorical clustering benchmarks. We demonstrate that the scaling of GALILEO is l…
We present a nonparametric Bayesian joint model for multivariate continuous and categorical variables, with the intention of developing a flexible engine for multiple imputation of missing values. The model fuses Dirichlet process mixtures of multinomial distributions for categorical variables with Dirichlet process mixtures of multivariate normal distributions for continuous variables. We incorporate dependence between the continuous and categorical variables by (i) modeling the means of the normal distributions as component-specific functions of the categorical variables and (ii) forming dis…
In this research, we estimate that around $27.99(\pm2.15)\%$ of the population has experienced wheezing before turning 1 in the United Kingdom. Furthermore, the Bernoulli Mixture Model classification is found to work best with $K=4$ clusters in order to better balance the separability of the clusters with their explanatory nature, based on a cohort of $N=1184$. The probability of the group of parents in the $j$th cluster to say that their children have wheezed during the $i$th age is assumed $P_{ij} \sim \text{Beta}(1/2, 1/2)$, the probabilities of assignment to each cluster is $R \sim \text{D…
In this study, we consider unsupervised clustering of categorical vectors that can be of different size using mixture. We use likelihood maximization to estimate the parameters of the underlying mixture model and a penalization technique to select the number of mixture components. Regardless of the true distribution that generated the data, we show that an explicit penalty, known up to a multiplicative constant, leads to a non-asymptotic oracle inequality with the Kullback-Leibler divergence on the two sides of the inequality. This theoretical result is illustrated by a document clustering app…
University of Waterloo（滑铁卢大学）；Vector Institute（向量研究所）；Noah’s Ark Lab（诺亚方舟实验室）；Huawei（华为）
The expectation-maximization (EM) algorithm has been widely used in minimizing the negative log likelihood (also known as cross entropy) of mixture models. However, little is understood about the goodness of the fixed points it converges to. In this paper, we study the regions where one component is missing in two-component mixture models, which we call one-cluster regions. We analyze the propensity of such regions to trap EM and gradient descent (GD) for mixtures of two Gaussians and mixtures of two Bernoullis. In the case of Gaussian mixtures, EM escapes one-cluster regions exponentially fas…
Multiple imputation is a common approach for dealing with missing values in statistical databases. The imputer fills in missing values with draws from predictive models estimated from the observed data, resulting in multiple, completed versions of the database. Researchers have developed a variety of default routines to implement multiple imputation; however, there has been limited research comparing the performance of these methods, particularly for categorical data. We use simulation studies to compare repeated sampling properties of three default multiple imputation methods for categorical …
We present a Bayesian model for estimating the joint distribution of multivariate categorical data when units are nested within groups. Such data arise frequently in social science settings, for example, people living in households. The model assumes that (i) each group is a member of a group-level latent class, and (ii) each unit is a member of a unit-level latent class nested within its group-level latent class. This structure allows the model to capture dependence among units in the same group. It also facilitates simultaneous modeling of variables at both group and unit levels. We develop …
We propose Dirichlet Process mixtures of Generalized Linear Models (DP-GLM), a new method of nonparametric regression that accommodates continuous and categorical inputs, and responses that can be modeled by a generalized linear model. We prove conditions for the asymptotic unbiasedness of the DP-GLM regression mean function estimate. We also give examples for when those conditions hold, including models for compactly supported continuous distributions and a model with continuous covariates and categorical response. We empirically analyze the properties of the DP-GLM and why it provides better…
Estimating the duration of user behavior is a central concern for most internet companies. Survival analysis is a promising method for analyzing the expected duration of events and usually assumes the same survival function for all subjects and the event will occur in the long run. However, such assumptions are inappropriate when the users behave differently or some events never occur for some users, i.e., the conversion period on web services of the light users with no intention of behaving actively on the service. Especially, if the proportion of inactive users is high, this assumption can l…
We propose a parsimonious extension of the classical latent class model to cluster categorical data by relaxing the class conditional independence assumption. Under this new mixture model, named Conditional Modes Model, variables are grouped into conditionally independent blocks. The corresponding block distribution is a parsimonious multinomial distribution where the few free parameters correspond to the most likely modality crossings, while the remaining probability mass is uniformly spread over the other modality crossings. Thus, the proposed model allows to bring out the intra-class depend…
The Adversarially Learned Mixture Model (AMM) is a generative model for unsupervised or semi-supervised data clustering. The AMM is the first adversarially optimized method to model the conditional dependence between inferred continuous and categorical latent variables. Experiments on the MNIST and SVHN datasets show that the AMM allows for semantic separation of complex data when little or no labeled data is available. The AMM achieves a state-of-the-art unsupervised clustering error rate of 2.86% on the MNIST dataset. A semi-supervised extension of the AMM yields competitive results on the S…
Statistical network analysis primarily focuses on inferring the parameters of an observed network. In many applications, especially in the social sciences, the observed data is the groups formed by individual subjects. In these applications, the network is itself a parameter of a statistical model. Zhao and Weko (2019) propose a model-based approach, called the hub model, to infer implicit networks from grouping behavior. The hub model assumes that each member of the group is brought together by a member of the group called the hub. The hub model belongs to the family of Bernoulli mixture mode…
Duke University（杜克大学）；University of Minnesota-Twin Cities（明尼苏达大学双城分校）
Motivated by the ever-increasing demands for limited communication bandwidth and low-power consumption, we propose a new methodology, named joint Variational Autoencoders with Bernoulli mixture models (VAB), for performing clustering in the compressed data domain. The idea is to reduce the data dimension by Variational Autoencoders (VAEs) and group data representations by Bernoulli mixture models (BMMs). Once jointly trained for compression and clustering, the model can be decomposed into two parts: a data vendor that encodes the raw data into compressed data, and a data consumer that classifi…
We propose a Bayesian nonparametric mixture model for prediction- and information extraction tasks with an efficient inference scheme. It models categorical-valued time series that exhibit dynamics from multiple underlying patterns (e.g. user behavior traces). We simplify the idea of capturing these patterns by hierarchical hidden Markov models (HHMMs) - and extend the existing approaches by the additional representation of structural information. Our empirical results are based on both synthetic- and real world data. They indicate that the results are easily interpretable, and that the model …
Multivariate categorical data are common in many fields. We are motivated by election polls studies assessing evidence of changes in voters opinions with their candidates preferences in the 2016 United States Presidential primaries or caucuses. Similar goals arise routinely in several applications, but current literature lacks a general methodology which combines flexibility, efficiency, and tractability in testing for group differences in multivariate categorical data at different---potentially complex---scales. We address this goal by leveraging a Bayesian representation which factorizes the…
Research on cluster analysis for categorical data continues to develop, with new clustering algorithms being proposed. However, in this context, the determination of the number of clusters is rarely addressed. In this paper, we propose a new approach in which clustering of categorical data and the estimation of the number of clusters is carried out simultaneously. Assuming that the data originate from a finite mixture of multinomial distributions, we develop a method to select the number of mixture components based on a minimum message length (MML) criterion and implement a new expectation-max…
Instituto Paulo Gontijo（保罗·贡蒂霍研究所）；FGV/EAESP（巴西应用经济研究所（FGV/EAESP））
In this work we cast the problem of binary classification in terms of estimating a partition on Bernoulli data. When the explanatory variables are all categorical, the problem can be modelled using the language of boolean functions. We offer a probabilistic analysis of the problem, and propose two algorithms for learning boolean functions from binary data.
PReMiuM is a recently developed R package for Bayesian clustering using a Dirichlet process mixture model. This model is an alternative to regression models, non-parametrically linking a response vector to covariate data through cluster membership. The package allows Bernoulli, Binomial, Poisson, Normal and categorical response, as well as Normal and discrete covariates. Additionally, predictions may be made for the response, and missing values for the covariates are handled. Several samplers and label switching moves are implemented along with diagnostic tools to assess convergence. A number …
Sequence analysis is being more and more widely used for the analysis of social sequences and other multivariate categorical time series data. However, it is often complex to describe, visualize, and compare large sequence data, especially when there are multiple parallel sequences per subject. Hidden (latent) Markov models (HMMs) are able to detect underlying latent structures and they can be used in various longitudinal settings: to account for measurement error, to detect unobservable states, or to compress information across several types of observations. Extending to mixture hidden Markov…
Helsinki Institute for Information Technology HIIT（赫尔辛基信息技术研究所（HIIT））；University of Helsinki（赫尔辛基大学）
Variational approximations are increasingly based on gradient-based optimization of expectations estimated by sampling. Handling discrete latent variables is then challenging because the sampling process is not differentiable. Continuous relaxations, such as the Gumbel-Softmax for categorical distribution, enable gradient-based optimization, but do not define a valid probability mass for discrete observations. In practice, selecting the amount of relaxation is difficult and one needs to optimize an objective that does not align with the desired one, causing problems especially with models havi…
We study Bayesian estimation of mixture models and argue in favor of fitting the marginal posterior distribution over component assignments directly, rather than Gibbs sampling from the joint posterior on components and parameters as is commonly done. Some previous authors have found the former approach to have slow mixing, but we show that, implemented correctly, it can achieve excellent performance. In particular, we describe a new Monte Carlo algorithm for sampling from the marginal posterior of a general integrable mixture that makes use of rejection-free sampling from the prior over compo…
Dasgupta and Shulman showed that a two-round variant of the EM algorithm can learn mixture of Gaussian distributions with near optimal precision with high probability if the Gaussian distributions are well separated and if the dimension is sufficiently high. In this paper, we generalize their theory to learning mixture of high-dimensional Bernoulli templates. Each template is a binary vector, and a template generates examples by randomly switching its binary components independently with a certain probability. In computer vision applications, a binary vector is a feature map of an image, where…
Continuous mixtures of distributions are widely employed in the statistical literature as models for phenomena with highly divergent outcomes; in particular, many familiar heavy-tailed distributions arise naturally as mixtures of light-tailed distributions (e.g., Gaussians), and play an important role in applications as diverse as modeling of extreme values and robust inference. In the case of social networks, continuous mixtures of graph distributions can likewise be employed to model social processes with heterogeneous outcomes, or as robust priors for network inference. Here, we introduce s…
Item nonresponse is frequently encountered in practice. Ignoring missing data can lose efficiency and lead to misleading inference. Fractional imputation is a frequentist approach of imputation for handling missing data. However, the parametric fractional imputation of \cite{kim2011parametric} may be subject to bias under model misspecification. In this paper, we propose a novel semiparametric fractional imputation method using Gaussian mixture models. The proposed method is computationally efficient and leads to robust estimation. The proposed method is further extended to incorporate the cat…
We present a Bayesian mixture model for estimating the joint distribution of mixed ordinal, nominal, and continuous data conditional on a set of fixed variables. The model uses multivariate normal and categorical mixture kernels for the random variables. It induces dependence between the random and fixed variables through the means of the multivariate normal mixture kernels and via a truncated local Dirichlet process. The latter encourages observations with similar values of the fixed variables to share mixture components. Using a simulation of data fusion, we illustrate that the model can est…
Identification of latent binary sequences from a pool of noisy observations has a wide range of applications in both statistical learning and population genetics. Each observed sequence is the result of passing one of the latent mother-sequences through a binary symmetric channel, which makes this configuration analogous to a special case of Bernoulli Mixture Models. This paper aims to attain an asymptotically tight upper-bound on the error of Maximum Likelihood mixture identification in such problems. The obtained results demonstrate fundamental guarantees on the inference accuracy of the opt…
In this paper, we are concerned with obtaining distribution-free concentration inequalities for mixture of independent Bernoulli variables that incorporate a notion of variance. Missing mass is the total probability mass associated to the outcomes that have not been seen in a given sample which is an important quantity that connects density estimates obtained from a sample to the population for discrete distributions. Therefore, we are specifically motivated to apply our method to study the concentration of missing mass - which can be expressed as a mixture of Bernoulli - in a novel way. We no…
Many panel studies collect refreshment samples---new, randomly sampled respondents who complete the questionnaire at the same time as a subsequent wave of the panel. With appropriate modeling, these samples can be leveraged to correct inferences for biases caused by non-ignorable attrition. We present such a model when the panel includes many categorical survey variables. The model relies on a Bayesian latent pattern mixture model, in which an indicator for attrition and the survey variables are modeled jointly via a latent class model. We allow the multinomial probabilities within classes to …
Choosing the number of mixture components remains an elusive challenge. Model selection criteria can be either overly liberal or conservative and return poorly-separated components of limited practical use. We formalize non-local priors (NLPs) for mixtures and show how they lead to well-separated components with non-negligible weight, interpretable as distinct subpopulations. We also propose an estimator for posterior model probabilities under local and non-local priors, showing that Bayes factors are ratios of posterior to prior empty-cluster probabilities. The estimator is widely applicable …
We propose a novel method for multiple clustering that assumes a co-clustering structure (partitions in both rows and columns of the data matrix) in each view. The new method is applicable to high-dimensional data. It is based on a nonparametric Bayesian approach in which the number of views and the number of feature-/subject clusters are inferred in a data-driven manner. We simultaneously model different distribution families, such as Gaussian, Poisson, and multinomial distributions in each cluster block. This makes our method applicable to datasets consisting of both numerical and categorica…
Standard linear modeling approaches make potentially simplistic assumptions regarding the structure of categorical effects that may obfuscate more complex relationships governing data. For example, recent work focused on the two-way unreplicated layout has shown that hidden groupings among the levels of one categorical predictor frequently interact with the ungrouped factor. We extend the notion of a "latent grouping factor" to linear models in general. The proposed work allows researchers to determine whether an apparent grouping of the levels of a categorical predictor reveals a plausible hi…
This paper considers the problem of mismeasured categorical covariates in the context of regression modeling; if unaccounted for, such misclassification is known to result in misestimation of model parameters. Here, we exploit the fact that explicitly modeling covariate misclassification leads to a mixture representation. Assuming common parametric families for the mixture components, and assuming that the misclassification occurrence is independent of the response variable, the mixture representation permits model parameters to be identified even when misclassification probabilities are unkno…
Multi-objective Bayesian Optimization With Mixed-categorical Design Variables for Expensive-to-evaluate Aeronautical Applications
ONERA（法国航空航天研究院）；Université de Toulouse（图卢兹大学）；ISAE–SUPAERO（法国高等航空航天学院）；INSA（法国国立应用科学学院）；CNRS（法国国家科学研究中心）；MINES ALBI（阿尔比矿业学校）；UPS Université de Toulouse（图卢兹第三大学）；Polytechnique Montréal（蒙特利尔理工学院）；German Aerospace Center (DLR)（德国航空航天中心）；EMBRAER S.A.（巴西航空工业公司）；University di Napoli Federico II（那不勒斯费德里科二世大学）
University of Liverpool（利物浦大学）；Universidad Antonio de Nebrija（安东尼奥·德·内布里哈大学）；Chalmers University of Technology（查尔姆斯理工大学）
This paper provides a comparative analysis between the adaptive birth model used in the labelled random finite set literature and the track initiation in the Poisson multi-Bernoulli mixture (PMBM) filter, with point-target models. The PMBM track initiation is obtained via Bayes' rule applied on the predicted PMBM density, and creates one Bernoulli component for each received measurement, representing that this measurement may be clutter or a detection from a new target. Adaptive birth mimics this procedure by creating a Bernoulli component for each measurement using a different rule to determi…
Pamminger and Fruwirth-Schnatter (2010) considered a Bayesian approach to model-based clustering of categorical time series assuming a fixed number of clusters. But the popular methods for selecting the number of clusters, for example, the Bayes Information Criterion (BIC), turned out to have severe problems in the categorical time series context. In this paper, we circumvent the difficulties of choosing the number of clusters by adopting the Bayesian semiparametric mixture model approach introduced by Bhattacharya (2008), who assume that the number of clusters is a random quantity, but is bou…
