Dirichlet-multinomial (DMN) distribution is commonly used to model over-dispersion in count data. Precise and fast numerical computation of the DMN log-likelihood function is important for performing statistical inference using this distribution, and remains a challenge. To address this, we use mathematical properties of the gamma function to derive a closed form expression for the DMN log-likelihood function. Compared to existing methods, calculation of the closed form has a lower computational complexity, hence is much faster without comprimising computational accuracy.
The multinomial and related distributions have long been used to model categorical, count-based data in fields ranging from bioinformatics to natural language processing. Commonly utilized variants include the standard multinomial and the Dirichlet multinomial distributions due to their computational efficiency and straightforward parameter estimation process. However, these distributions make strict assumptions about the mean, variance, and covariance between the categorical features being modeled. If these assumptions are not met by the data, it may result in poor parameter estimates and los…
For a multinomial distribution, suppose that we have prior knowledge of the sum of the probabilities of some categories. This allows us to construct a submodel in a full (i.e., no-restriction) model. Maximum likelihood estimation (MLE) under this submodel is expected to have better estimation efficiency than MLE under the full model. This article presents the asymptotic expansion of the risk of MLE with respect to Kullback--Leibler divergence for both the full model and submodel. The results reveal that, using the submodel, the reduction of the risk is quite small in some cases. Furthermore, w…
Traditionally, the Dirichlet-multinomial distribution has been recognized as a key model for contingency tables generated by cluster sampling schemes. There are, however, other possible distributions appropriate for these contingency tables. This paper introduces new test-statistics capable to test log-linear modeling hypotheses with no distributional specification, when the individuals of the clusters are possibly homogeneously correlated. The estimator for the intracluster correlation coefficient proposed in Alonso-Revenga et al. (2016), valid for different cluster sizes, plays a crucial rol…
In Quantum Non Demolition measurements, the sequence of observations is distributed as a mixture of multinomial random variables. Parameters of the dynamics are naturally encoded into this family of distributions. We show the local asymptotic mixed normality of the underlying statistical model and the consistency of the maximum likelihood estimator. Furthermore, we prove the asymptotic optimality of this estimator as it saturates the usual Cramér Rao bound.
We consider a problem of clustering a sequence of multinomial observations by way of a model selection criterion. We propose a form of a penalty term for the model selection procedure. Our approach subsumes both the conventional AIC and BIC criteria but also extends the conventional criteria in a way that it can be applicable also to a sequence of sparse multinomial observations, where even within a same cluster, the number of multinomial trials may be different for different observations. In addition, as a preliminary estimation step to maximum likelihood estimation, and more generally, to ma…
We study the problem of maximum likelihood estimation for general patterns of bivariate missing data for normal and multinomial random variables, under the assumption that the data is missing at random (MAR). For normal data, the score equations have nine complex solutions, at least one of which is real and statistically significant. Our computations suggest that the number of real solutions is related to whether or not the MAR assumption is satisfied. In the multinomial case, all solutions to the score equations are real and the number of real solutions grows exponentially in the number of st…
Estimation of a single Bernoulli parameter using pooled sampling is among the oldest problems in the group testing literature. To carry out such estimation, an array of efficient estimators have been introduced covering a wide range of situations routinely encountered in applications. More recently, there has been growing interest in using group testing to simultaneously estimate the joint probabilities of two correlated traits using a multinomial model. Unfortunately, basic estimation results, such as the maximum likelihood estimator (MLE), have not been adequately addressed in the literature…
The statistical analysis of discrete data has been the subject of extensive statistical research dating back to the work of Pearson. In this survey we review some recently developed methods for testing hypotheses about high-dimensional multinomials. Traditional tests like the $χ^2$ test and the likelihood ratio test can have poor power in the high-dimensional setting. Much of the research in this area has focused on finding tests with asymptotically Normal limits and developing (stringent) conditions under which tests have Normal limits. We argue that this perspective suffers from a significan…
A General Theory for Softmax Gating Multinomial Logistic Mixture of Experts
The University of Texas at Austin（德克萨斯大学奥斯汀分校）；Univ. Grenoble Alpes（格勒诺布尔大学）；Inria（法国国家信息与自动化研究所）；CNRS（法国国家科学研究中心）；Grenoble INP（格勒诺布尔理工学院）；LJK（约瑟夫·傅里叶实验室）；The University of Queensland（昆士兰大学）
The object of study is the problem of testing for uniformity of the multinomial distribution. We consider tests based on symmetric statistics, defined as the sum of some function of cell-frequencies. Mainly, attention is focused on the class of power divergence statistics, in particular, on the chi-square and log-likelihood ratio statistics. The main issue of the article is to study the asymptotic properties of tests at the concept of an intermediate setting in terms of so called -intermediate asymptotic efficiency due to Ivchenko and Mirakhmedov (1995), when the asymptotic power of tests are …
We introduce a semiparametric ``tubular neighborhood'' of a parametric model in the multinomial setting. It consists of all multinomial distributions lying in a distance-based neighborhood of the parametric model of interest. Fitting such a tubular model allows one to use a parametric model while treating it as an approximation to the true distribution. In this paper, the Kullback--Leibler distance is used to build the tubular region. Based on this idea one can define the distance between the true multinomial distribution and the parametric model to be the index of fit. The paper develops a li…
The Dirichlet distribution, also known as multivariate beta, is the most used to analyse frequencies or proportions data. Maximum likelihood is widespread for estimation of Dirichlet's parameters. However, for small sample sizes, the maximum likelihood estimator may shows a significant bias. In this paper, Dirchlet's parameters estimation is obtained through modified score functions aiming at mean and median bias reduction of the maximum likelihood estimator, respectively. A simulation study and an application compare the adjusted score approaches with maximum likelihood.
In this paper we consider Poisson loglinear models with linear constraints (LMLC) on the expected table counts. Multinomial and product multinomial loglinear models can be obtained by considering that some marginal totals (linear constraints on the expected table counts) have been prefixed in a Poisson loglinear model. Therefore with the theory developed in this paper, multinomial and product multinomial loglinear models can be considered as a particular case. To carry out inferences on the parameters in the LMLC an information-theoretic approach is followed from which the classical maximum li…
We study shrinkage estimation of the mean parameters of a class of multivariate distributions for which the diagonal entries of the corresponding covariance matrix are certain quadratic functions of the mean parameter. This class of distributions includes the diagonal multivariate natural exponential families. We propose two classes of semi-parametric shrinkage estimators for the mean and construct unbiased estimators of the corresponding risk. We establish the asymptotic consistency and convergence rates for these shrinkage estimators under squared error loss as both $n$, the sample size, and…
Fast Model Selection and Stable Optimization for Softmax-Gated Multinomial-Logistic Mixture of Experts Models
School of Computing, National University of Singapore, Singapore.（新加坡国立大学计算机学院）；ARC Centre of Excellence for the Mathematical Analysis of Cellular Systems（细胞系统数学分析卓越研究中心）；School of Mathematical Sciences, Queensland University of Technology, Brisbane, Australia.（昆士兰科技大学数学科学学院）；Department of Statistics and Data Science, University of Texas at Austin, Austin, USA（德克萨斯大学奥斯汀分校统计与数据科学系）
Learning in Position-Aware Multinomial Logit Bandits: From Multiplicative to General Position Effects
Leonard N. Stern School of Business, New York University（纽约大学勒纳商学院）；Qiuzhen College, Tsinghua University（清华大学齐臻学院）；Department of Management Science, School of Management, Fudan University（复旦大学管理学院管理科学系）；Yau Mathematical Sciences Center & Department of Mathematical Sciences, Tsinghua University（清华大学尤数学科学中心及数学科学系）
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.
Recently Liu and Wang derived the likelihood ratio test (LRT) statistic and its asymptotic distribution for testing equality of two multinomial distributions vs. the alternative that the second distribution is larger in terms of increasing convex order (ICX). ICX is less restrictive than stochastic order and is a notion that has found applications in insurance and actuarial science. In this paper we propose a new test for ICX. The new test has several advantages over the LRT and over any test procedure that depends on asymptotic theory for implementation. The advantages include the following: …
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））
When estimating a proportion and only a sample of triplets is given, dependencies within the triplets are to be accounted for. Without assuming a distribution for the success count of the triplet, together with the proportion, as second and third parameter the correlations of 1st and 2nd order enter the model. We apply maximum likelihood estimation, and derive consistency by using that the triplet count is multinomially distributed, combined with the continuous mapping theorem. The asymptotic normality follows with the delta-method, resulting in closed-form expressions for the standard errors.…
The MLE (Maximum Likelihood Estimate) for a multinomial model is proportional to the data. We call such estimate an eigenestimate and the relationship of it to the data as the eigenstructure. When the multinomial model is generalized to deal with data arise from incomplete or censored categorical counts, we would naturally look for this eigenstructure between MLE and data. The paper finds the algebraic representation of the eigenstructure (put as Eqn (2.1)), with which the intuition is visualized geometrically (Figures 2.2 and 4.3) and elaborated in a theory (Section 4). The eigenestimate cons…
The paper considers general multiplicative models for complete and incomplete contingency tables that generalize log-linear and several other models and are entirely coordinate free. Sufficient conditions of the existence of maximum likelihood estimates under these models are given, and it is shown that the usual equivalence between multinomial and Poisson likelihoods holds if and only if an overall effect is present in the model. If such an effect is not assumed, the model becomes a curved exponential family and a related mixed parameterization is given that relies on non-homogeneous odds rat…
We show that the mean-model parameter is always orthogonal to the error distribution in generalized linear models. Thus, the maximum likelihood estimator of the mean-model parameter will be asymptotically efficient regardless of whether the error distribution is known completely, known up to a finite vector of parameters, or left completely unspecified, in which case the likelihood is taken to be an appropriate semiparametric likelihood. Moreover, the maximum likelihood estimator of the mean-model parameter will be asymptotically independent of the maximum likelihood estimator of the error dis…
In latent Dirichlet allocation (LDA), topics are multinomial distributions over the entire vocabulary. However, the vocabulary usually contains many words that are not relevant in forming the topics. We adopt a variable selection method widely used in statistical modeling as a dimension reduction tool and combine it with LDA. In this variable selection model for LDA (vsLDA), topics are multinomial distributions over a subset of the vocabulary, and by excluding words that are not informative for finding the latent topic structure of the corpus, vsLDA finds topics that are more robust and discri…
Likelihood ratio tests are widely used to test statistical hypotheses about parametric families of probability distributions. If interest is restricted to a subfamily of distributions, then it is natural to inquire if the restricted LRT is superior to the unrestricted LRT. Marden's general LRT conjecture posits that any restriction placed on the alternative hypothesis will increase power. The only published counterexample to this conjecture is rather technical and involves a restriction that maintains the dimension of the alternative. We formulate the dimension-restricted LRT conjecture, which…
We consider probabilistic multinomial probit classification using Gaussian process (GP) priors. The challenges with the multiclass GP classification are the integration over the non-Gaussian posterior distribution, and the increase of the number of unknown latent variables as the number of target classes grows. Expectation propagation (EP) has proven to be a very accurate method for approximate inference but the existing EP approaches for the multinomial probit GP classification rely on numerical quadratures or independence assumptions between the latent values from different classes to facili…
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））
For the estimation of cumulative link models for ordinal data, the bias-reducing adjusted score equations in \citet{firth:93} are obtained, whose solution ensures an estimator with smaller asymptotic bias than the maximum likelihood estimator. Their form suggests a parameter-dependent adjustment of the multinomial counts, which, in turn suggests the solution of the adjusted score equations through iterated maximum likelihood fits on adjusted counts, greatly facilitating implementation. Like the maximum likelihood estimator, the reduced-bias estimator is found to respect the invariance properti…
Relational models generalize log-linear models to arbitrary discrete sample spaces by specifying effects associated with any subsets of their cells. A relational model may include an overall effect, pertaining to every cell after a reparameterization, and in this case, the properties of the maximum likelihood estimates (MLEs) are analogous to those computed under traditional log-linear models, and the goodness-of-fit tests are also the same. If an overall effect is not present in any reparameterization, the properties of the MLEs are considerably different, and the Poisson and multinomial MLEs…
Handling Missingness and Censoring in Dirichlet Models
University of Pretoria（比勒陀利亚大学）；National Institute for Theoretical and Computational Sciences (NITheCS)（国家理论与计算科学研究所）；San José State University（圣何塞州立大学）；University of Catania（卡塔尼亚大学）
Missingness in categorical data is a common problem in various real applications. Traditional approaches either utilize only the complete observations or impute the missing data by some ad hoc methods rather than the true conditional distribution of the missing data, thus losing or distorting the rich information in the partial observations. In this paper, we propose the Dirichlet Process Mixture of Collapsed Product-Multinomials (DPMCPM) to model the full data jointly and compute the model efficiently. By fitting an infinite mixture of product-multinomial distributions, DPMCPM is applicable f…
School of Mathematics, University of Bristol（布里斯托大学数学学院）；the Alan Turing Institute（阿兰·图灵研究所）
We introduce a new method for inference in stochastic epidemic models which uses recursive multinomial approximations to integrate over unobserved variables and thus circumvent likelihood intractability. The method is applicable to a class of discrete-time, finite-population compartmental models with partial, randomly under-reported or missing count observations. In contrast to state-of-the-art alternatives such as Approximate Bayesian Computation techniques, no forward simulation of the model is required and there are no tuning parameters. Evaluating the approximate marginal likelihood of mod…
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…
We propose a penalized likelihood method that simultaneously fits the multinomial logistic regression model and combines subsets of the response categories. The penalty is non differentiable when pairs of columns in the optimization variable are equal. This encourages pairwise equality of these columns in the estimator, which corresponds to response category combination. We use an alternating direction method of multipliers algorithm to compute the estimator and we discuss the algorithm's convergence. Prediction and model selection are also addressed.
University of Amsterdam（阿姆斯特丹大学）；Technical University of Denmark（丹麦技术大学）
Generative flows and diffusion models have been predominantly trained on ordinal data, for example natural images. This paper introduces two extensions of flows and diffusion for categorical data such as language or image segmentation: Argmax Flows and Multinomial Diffusion. Argmax Flows are defined by a composition of a continuous distribution (such as a normalizing flow), and an argmax function. To optimize this model, we learn a probabilistic inverse for the argmax that lifts the categorical data to a continuous space. Multinomial Diffusion gradually adds categorical noise in a diffusion pr…
We extend variational autoencoders (VAEs) to collaborative filtering for implicit feedback. This non-linear probabilistic model enables us to go beyond the limited modeling capacity of linear factor models which still largely dominate collaborative filtering research.We introduce a generative model with multinomial likelihood and use Bayesian inference for parameter estimation. Despite widespread use in language modeling and economics, the multinomial likelihood receives less attention in the recommender systems literature. We introduce a different regularization parameter for the learning obj…
Bayesian neural networks (BNNs) can account for both aleatoric and epistemic uncertainty. However, in BNNs the priors are often specified over the weights which rarely reflects true prior knowledge in large and complex neural network architectures. We present a simple approach to incorporate prior knowledge in BNNs based on external summary information about the predicted classification probabilities for a given dataset. The available summary information is incorporated as augmented data and modeled with a Dirichlet process, and we derive the corresponding \emph{Summary Evidence Lower BOund}. …
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…
Count data take on non-negative integer values and are challenging to properly analyze using standard linear-Gaussian methods such as linear regression and principal components analysis. Generalized linear models enable direct modeling of counts in a regression context using distributions such as the Poisson and negative binomial. When counts contain only relative information, multinomial or Dirichlet-multinomial models can be more appropriate. We review some of the fundamental connections between multinomial and count models from probability theory, providing detailed proofs. These relationsh…
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 derive the conjugate prior of the Dirichlet and beta distributions and explore it with numerical examples to gain an intuitive understanding of the distribution itself, its hyperparameters, and conditions concerning its convergence. Due to the prior's intractability, we proceed to define and analyze a closed-form approximation. Finally, we provide an algorithm implementing this approximation that enables fully tractable Bayesian conjugate treatment of Dirichlet and beta likelihoods without the need for Monte Carlo simulations.
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…
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…
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…
