Studying the human microbiome has gained substantial interest in recent years, and a common task in the analysis of these data is to cluster microbiome compositions into subtypes. This subdivision of samples into subgroups serves as an intermediary step in achieving personalized diagnosis and treatment. In applying existing clustering methods to modern microbiome studies including the American Gut Project (AGP) data, we found that this seemingly standard task, however, is very challenging in the microbiome composition context due to several key features of such data. Standard distance-based clustering algorithms generally do not produce reliable results as they do not take into account the heterogeneity of the cross-sample variability among the bacterial taxa, while existing model-based approaches do not allow sufficient flexibility for the identification of complex within-cluster variation from cross-cluster variation. Direct applications of such methods generally lead to overly dispersed clusters in the AGP data and such phenomenon is common for other microbiome data. To overcome these challenges, we introduce Dirichlet-tree multinomial mixtures (DTMM) as a Bayesian generative model for clustering amplicon sequencing data in microbiome studies. DTMM models the microbiome population with a mixture of Dirichlet-tree kernels that utilizes the phylogenetic tree to offer a more flexible covariance structure in characterizing within-cluster variation, and it provides a means for identifying a subset of signature taxa that distinguish the clusters. We perform extensive simulation studies to evaluate the performance of DTMM and compare it to state-of-the-art model-based and distance-based clustering methods in the microbiome context. Finally, we report a case study on the fecal data from the AGP to identify compositional clusters among individuals with inflammatory bowel disease and diabetes.
The human microbiome plays an important role in human health and disease status. Next generating sequencing technologies allow for quantifying the composition of the human microbiome. Clustering these microbiome data can provide valuable information by identifying underlying patterns across samples. Recently, Fang and Subedi (2020) proposed a logistic normal multinomial mixture model (LNM-MM) for clustering microbiome data. As microbiome data tends to be high dimensional, here, we develop a family of logistic normal multinomial factor analyzers (LNM-FA) by incorporating a factor analyzer struc…
Discrete data such as counts of microbiome taxa resulting from next-generation sequencing are routinely encountered in bioinformatics. Taxa count data in microbiome studies are typically high-dimensional, over-dispersed, and can only reveal relative abundance therefore being treated as compositional. Analyzing compositional data presents many challenges because they are restricted on a simplex. In a logistic normal multinomial model, the relative abundance is mapped from a simplex to a latent variable that exists on the real Euclidean space using the additive log-ratio transformation. While a …
High-throughput sequencing technology provides unprecedented opportunities to quantitatively explore human gut microbiome and its relation to diseases. Microbiome data are compositional, sparse, noisy, and heterogeneous, which pose serious challenges for statistical modeling. We propose an identifiable Bayesian multinomial matrix factorization model to infer overlapping clusters on both microbes and hosts. The proposed method represents the observed over-dispersed zero-inflated count matrix as Dirichlet-multinomial mixtures on which latent cluster structures are built hierarchically. Under the…
There is a keen interest in characterizing variation in the microbiome across cancer patients, given increasing evidence of its important role in determining treatment outcomes. Here our goal is to discover subgroups of patients with similar microbiome profiles. We propose a novel unsupervised clustering approach in the Bayesian framework that innovates over existing model-based clustering approaches, such as the Dirichlet multinomial mixture model, in three key respects: we incorporate feature selection, learn the appropriate number of clusters from the data, and integrate information on the …
This paper develops a Bayesian graphical model for fusing disparate types of count data. The motivating application is the study of bacterial communities from diverse high dimensional features, in this case transcripts, collected from different treatments. In such datasets, there are no explicit correspondences between the communities and each correspond to different factors, making data fusion challenging. We introduce a flexible multinomial-Gaussian generative model for jointly modeling such count data. This latent variable model jointly characterizes the observed data through a common multi…
We consider the problem of drawing samples from posterior distributions formed under a Dirichlet prior and a truncated multinomial likelihood, by which we mean a Multinomial likelihood function where we condition on one or more counts being zero a priori. Sampling this posterior distribution is of interest in inference algorithms for hierarchical Bayesian models based on the Dirichlet distribution or the Dirichlet process, particularly Gibbs sampling algorithms for the Hierarchical Dirichlet Process Hidden Semi-Markov Model. We provide a data augmentation sampling algorithm that is easy to imp…
Mixed-membership (MM) models such as Latent Dirichlet Allocation (LDA) have been applied to microbiome compositional data to identify latent subcommunities of microbial species. These subcommunities are informative for understanding the biological interplay of microbes and for predicting health outcomes. However, microbiome compositions typically display substantial cross-sample heterogeneities in subcommunity compositions -- that is, the variability in the proportions of microbes in shared subcommunities across samples -- which is not accounted for in prior analyses. As a result, LDA can prod…
We describe a procedure to introduce general dependence structures on a set of Dirichlet processes. Dependence can be in one direction to define a time series or in two directions to define spatial dependencies. More directions can also be considered. Dependence is induced via a set of latent processes and exploit the conjugacy property between the Dirichlet and the multinomial processes to ensure that the marginal law for each element of the set is a Dirichlet process. Dependence is characterised through the correlation between any two elements. Posterior distributions are obtained when we us…
Bayesian multinomial logistic-normal (MLN) models are popular for the analysis of sequence count data (e.g., microbiome or gene expression data) due to their ability to model multivariate count data with complex covariance structure. However, existing implementations of MLN models are limited to handling small data sets due to the non-conjugacy of the multinomial and logistic-normal distributions. We introduce MLN models which can be written as marginally latent matrix-t process (LTP) models. Marginally LTP models describe a flexible class of generalized linear regression, non-linear regressio…
The Dirichlet-multinomial (DM) distribution plays a fundamental role in modern statistical methodology development and application. Recently, the DM distribution and its variants have been used extensively to model multivariate count data generated by high-throughput sequencing technology in omics research due to its ability to accommodate the compositional structure of the data as well as overdispersion. A major limitation of the DM distribution is that it is unable to handle excess zeros typically found in practice which may bias inference. To fill this gap, we propose a novel Bayesian zero-…
A reparametrized Dirichlet-multinomial distribution is introduced, and the covariance matrix, as well as, the algorithm for calculating the PDF for n species are provided. The distribution is suited for modelling the joint distribution of pin-point cover data of spatially aggregated plant species, and the parametrization ensures that the degree of spatially aggregation is modelled by a parameter and the mean cover of the species is modelled by the other parameters. This last property is convenient for using the distribution in Bayesian hierarchical models where the mean cover of the different …
In this paper, we discuss the construction of a multivariate generalisation of the Dirichlet-multinomial distribution. An example from forensic genetics in the statistical analysis of DNA mixtures motivates the study of this multivariate extension. In forensic genetics, adjustment of the match probabilities due to remote ancestry in the population is often done using the so-called θ-correction. This correction increases the probability of observing multiple copies of rare alleles and thereby reduces the weight of the evidence for rare genotypes. By numerical examples, we show how the θ-correct…
Detecting associations between microbial compositions and sample characteristics is one of the most important tasks in microbiome studies. Most of the existing methods apply univariate models to single microbial species separately, with adjustments for multiple hypothesis testing. We propose a Bayesian analysis for a generalized mixed effects linear model tailored to this application. The marginal prior on each microbial composition is a Dirichlet Process, and dependence across compositions is induced through a linear combination of individual covariates, such as disease biomarkers or the subj…
The human microbiome is a complex ecological system, and describing its structure and function under different environmental conditions is important from both basic scientific and medical perspectives. Viewed through a biostatistical lens, many microbiome analysis goals can be formulated as latent variable modeling problems. However, although probabilistic latent variable models are a cornerstone of modern unsupervised learning, they are rarely applied in the context of microbiome data analysis, in spite of the evolutionary, temporal, and count structure that could be directly incorporated thr…
We present a novel approach for estimating conditional probability tables, based on a joint, rather than independent, estimate of the conditional distributions belonging to the same table. We derive exact analytical expressions for the estimators and we analyse their properties both analytically and via simulation. We then apply this method to the estimation of parameters in a Bayesian network. Given the structure of the network, the proposed approach better estimates the joint distribution and significantly improves the classification performance with respect to traditional approaches.
This paper generalises the exponential family GLM to allow arbitrary distributions for the response variable. This is achieved by combining the model-assisted regression approach from survey sampling with the GLM scoring algorithm, weighted by random draws from the posterior Dirichlet distribution of the support point probabilities of the multinomial distribution. The generalisation provides fully Bayesian analyses from the posterior sampling, without MCMC. Several examples are given, of published GLM data sets. The approach can be extended widely: an example of a GLMM extension is given.
One-step ahead prediction for the multinomial model is considered. The performance of a predictive density is evaluated by the average Kullback-Leibler divergence from the true density to the predictive density. Asymptotic approximations of risk functions of Bayesian predictive densities based on Dirichlet priors are obtained. It is shown that a Bayesian predictive density based on a specific Dirichlet prior is asymptotically minimax. The asymptotically minimax prior is different from known objective priors such as the Jeffreys prior or the uniform prior.
Online estimation and modelling of i.i.d. data for short sequences over large or complex "alphabets" is a ubiquitous (sub)problem in machine learning, information theory, data compression, statistical language processing, and document analysis. The Dirichlet-Multinomial distribution (also called Polya urn scheme) and extensions thereof are widely applied for online i.i.d. estimation. Good a-priori choices for the parameters in this regime are difficult to obtain though. I derive an optimal adaptive choice for the main parameter via tight, data-dependent redundancy bounds for a related model. T…
We propose two Bayesian multinomial-Dirichlet models to predict the final outcome of football (soccer) matches and compare them to three well-known models regarding their predictive power. All the models predicted the full-time results of 1710 matches of the first division of the Brazilian football championship and the comparison used three proper scoring rules, the proportion of errors and a calibration assessment. We also provide a goodness of fit measure. Our results show that multinomial-Dirichlet models are not only competitive with standard approaches, but they are also well calibrated a…
Choose A Table: Tensor Dirichlet Process Multinomial Mixture Model with Graphs for Passenger Trajectory Clustering
University of Cologne（科隆大学）；Arizona State University（亚利桑那州立大学）；Tsinghua University（清华大学）；McGill University（麦吉尔大学）；The Hong Kong University of Science and Technology（香港科技大学）；HKUST-GZ（香港科技大学（广州））
Dirichlet distribution and Dirichlet process as its infinite dimensional generalization are primarily used conjugate prior of categorical and multinomial distributions in Bayesian statistics. Extensions have been proposed to broaden applications for different purposes. In this article, we explore a class of prior distributions closely related to Dirichlet distribution incorporating additional information on the data generating mechanism. Examples are given to show potential use of the models.
Dust particles recovered from the soles of shoes may be indicative of the sites recently visited by an individual, and, in particular, of the presence of an individual at a particular site of interest, e.g., the scene of a crime. By describing the dust profile of a given site by a multinomial distribution over a fixed number of dust particle types, we can define the probability distribution of the mixture of dust recovered from the sole of a shoe via Latent Dirichlet Allocation. We use Variational Bayesian Inference to study the parameters of the model, and use their resulting posterior distri…
This article focuses on the performance of Bayes estimators, in comparison with the MLE, in multinomial models with a relatively large number of cells. The prior for the Bayes estimator is taken to be the conjugate Dirichlet, i.e., the multivariate Beta, with exchangeable distributions over the coordinates, including the non-informative uniform distribution. The choice of the multinomial is motivated by its many applications in business and industry, but also by its use in providing a simple nonparametric estimator of an unknown distribution. It is striking that the Bayes procedure outperforms…
Tensor Dirichlet Process Multinomial Mixture Model for Passenger Trajectory Clustering
University of Cologne（科隆大学）；Arizona State University（亚利桑那州立大学）；Tsinghua University（清华大学）；McGill University（麦吉尔大学）；The Hong Kong University of Science and Technology（香港科技大学）；HKUST (Guangzhou Campus)（香港科技大学（广州校区））
Many practical modeling problems involve discrete data that are best represented as draws from multinomial or categorical distributions. For example, nucleotides in a DNA sequence, children's names in a given state and year, and text documents are all commonly modeled with multinomial distributions. In all of these cases, we expect some form of dependency between the draws: the nucleotide at one position in the DNA strand may depend on the preceding nucleotides, children's names are highly correlated from year to year, and topics in text may be correlated and dynamic. These dependencies are no…
Public sentiment is a direct public-centric indicator for the success of effective action planning. Despite its importance, systematic modeling of public sentiment remains untapped in previous studies. This research aims to develop a Bayesian-based approach for quantitative public sentiment modeling, which is capable of incorporating uncertainty and guiding the selection of public sentiment measures. This study comprises three steps: (1) quantifying prior sentiment information and new sentiment observations with Dirichlet distribution and multinomial distribution respectively; (2) deriving the…
University of Pretoria（比勒陀利亚大学）；Center for Artificial Intelligence Research (CAIR)（人工智能研究中心（CAIR））
Most topic models are constructed under the assumption that documents follow a multinomial distribution. The Poisson distribution is an alternative distribution to describe the probability of count data. For topic modelling, the Poisson distribution describes the number of occurrences of a word in documents of fixed length. The Poisson distribution has been successfully applied in text classification, but its application to topic modelling is not well documented, specifically in the context of a generative probabilistic model. Furthermore, the few Poisson topic models in literature are admixtu…
We present a mixed multinomial logit (MNL) model, which leverages the truncated stick-breaking process representation of the Dirichlet process as a flexible nonparametric mixing distribution. The proposed model is a Dirichlet process mixture model and accommodates discrete representations of heterogeneity, like a latent class MNL model. Yet, unlike a latent class MNL model, the proposed discrete choice model does not require the analyst to fix the number of mixture components prior to estimation, as the complexity of the discrete mixing distribution is inferred from the evidence. For posterior…
Hierarchical mixtures of Unigram models for short text clustering: The role of Beta-Liouville priors
Department of Precision and Regenerative Medicine and Jonian Area (DiMePRe-J)（精准与再生医学系及乔尼亚区）；University of Bari Aldo Moro（巴里阿尔多·莫罗大学）；Sirio Astronomical Observatory（西里奥天文台）；Grotte di Castellana srl（卡斯特尔纳纳洞窟有限公司）
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 technical report, we present jLDADMM---an easy-to-use Java toolkit for conventional topic models. jLDADMM is released to provide alternatives for topic modeling on normal or short texts. It provides implementations of the Latent Dirichlet Allocation topic model and the one-topic-per-document Dirichlet Multinomial Mixture model (i.e. mixture of unigrams), using collapsed Gibbs sampling. In addition, jLDADMM supplies a document clustering evaluation to compare topic models. jLDADMM is open-source and available to download at: https://github.com/datquocnguyen/jLDADMM
We use the theory of normal variance-mean mixtures to derive a data augmentation scheme for models that include gamma functions. Our methodology applies to many situations in statistics and machine learning, including Multinomial-Dirichlet distributions, Negative binomial regression, Poisson-Gamma hierarchical models, Extreme value models, to name but a few. All of those models include a gamma function which does not admit a natural conjugate prior distribution providing a significant challenge to inference and prediction. To provide a data augmentation strategy, we construct and develop the t…
Although fully generative models have been successfully used to model the contents of text documents, they are often awkward to apply to combinations of text data and document metadata. In this paper we propose a Dirichlet-multinomial regression (DMR) topic model that includes a log-linear prior on document-topic distributions that is a function of observed features of the document, such as author, publication venue, references, and dates. We show that by selecting appropriate features, DMR topic models can meet or exceed the performance of several previously published topic models designed fo…
The estimation of voting blocs is an important statistical inquiry in political science. However, the scope of these analyses is usually restricted to roll call data where individual votes are directly observed. Here, we examine a Bayesian mixture model with Dirichlet-multinomial components to infer voting blocs within longitudinal referendum data aggregated at the municipal level. As a case study, we analyze vote totals from 423 municipalities in the US state Maine for 54 referendum questions balloted from 2008-2019. Using this model, we recover the posterior distribution on the number of vot…
The large number of publicly available survey datasets of wide variety, albeit useful, raise respondent-level privacy concerns. The synthetic data approach to data privacy and confidentiality has been shown useful in terms of privacy protection and utility preservation. This paper aims at illustrating how synthetic data can facilitate the dissemination of highly sensitive information about youth risk behavior by presenting a case study of synthetic data for a sample of the Youth Risk Behavior Survey (YRBS). Given the categorical nature of almost all variables in YRBS, the Dirichlet Process mix…
Topic models, such as latent Dirichlet allocation (LDA), can be useful tools for the statistical analysis of document collections and other discrete data. The LDA model assumes that the words of each document arise from a mixture of topics, each of which is a distribution over the vocabulary. A limitation of LDA is the inability to model topic correlation even though, for example, a document about genetics is more likely to also be about disease than X-ray astronomy. This limitation stems from the use of the Dirichlet distribution to model the variability among the topic proportions. In this p…
Fossil-based palaeoclimate reconstruction is an important area of ecological science that has gained momentum in the backdrop of the global climate change debate. The hierarchical Bayesian paradigm provides an interesting platform for studying such important scientific issue. However, our cross-validation based assessment of the existing Bayesian hierarchical models with respect to two modern proxy data sets based on chironomid and pollen, respectively, revealed that the models are inadequate for the data sets. In this paper, we model the species assemblages (compositional data) by the zero-in…
Multivariate categorical data nested within households often include reported values that fail edit constraints---for example, a participating household reports a child's age as older than his biological parent's age---as well as missing values. Generally, agencies prefer datasets to be free from erroneous or missing values before analyzing them or disseminating them to secondary data users. We present a model-based engine for editing and imputation of household data based on a Bayesian hierarchical model that includes (i) a nested data Dirichlet process mixture of products of multinomial dist…
Feature selection represents a measure to reduce the complexity of high-dimensional datasets and gain insights into the systematic variation in the data. This aspect is of specific importance in domains that rely on model interpretability, such as life sciences. We propose UBayFS, an ensemble feature selection technique embedded in a Bayesian statistical framework. Our approach considers two sources of information: data and domain knowledge. We build a meta-model from an ensemble of elementary feature selectors and aggregate this information in a multinomial likelihood. The user guides UBayFS …
In immunological studies, the characterization of small, functionally distinct cell subsets from blood and tissue is crucial to decipher system level biological changes. An increasing number of studies rely on assays that provide single-cell measurements of multiple genes and proteins from bulk cell samples. A common problem in the analysis of such data is to identify biomarkers (or combinations of thereof) that are differentially expressed between two biological conditions (e.g., before/after vaccination), where expression is defined as the proportion of cells expressing the biomarker or comb…
Common election poll reporting is often misleading as sample uncertainty is addressed insufficiently or not covered at all. Furthermore, main interest usually lies beyond the simple party shares. For a more comprehensive opinion poll and election coverage, we propose shifting the focus towards the reporting of survey-based probabilities for specific events of interest. We present such an approach for multi-party electoral systems, focusing on probabilities of coalition majorities. A Monte Carlo approach based on a Bayesian Multinomial-Dirichlet model is used for estimation. Probabilities are e…
