As in other estimation scenarios, likelihood based estimation in the normal mixture set-up is highly non-robust against model misspecification and presence of outliers (apart from being an ill-posed optimization problem). A robust alternative to the ordinary likelihood approach for this estimation problem is proposed which performs simultaneous estimation and data clustering and leads to subsequent anomaly detection. To invoke robustness, the methodology based on the minimization of the density power divergence (or alternatively, the maximization of the $β$-likelihood) is utilized under suitable constraints. An iteratively reweighted least squares approach has been followed in order to compute the proposed estimators for the component means (or equivalently cluster centers) and component dispersion matrices which leads to simultaneous data clustering. Some exploratory techniques are also suggested for anomaly detection, a problem of great importance in the domain of statistics and machine learning. The proposed method is validated with simulation studies under different set-ups; it performs competitively or better compared to the popular existing methods like K-medoids, TCLUST, trimmed K-means and MCLUST, especially when the mixture components (i.e., the clusters) share regions with significant overlap or outlying clusters exist with small but non-negligible weights (particularly in higher dimensions). Two real datasets are also used to illustrate the performance of the newly proposed method in comparison with others along with an application in image processing. The proposed method detects the clusters with lower misclassification rates and successfully points out the outlying (anomalous) observations from these datasets.
Robust estimation under multivariate normal (MVN) mixture model is always a computational challenge. A recently proposed maximum pseudo \b{eta}-likelihood estimator aims to estimate the unknown parameters of a MVN mixture model in the spirit of minimum density power divergence (DPD) methodology but with a relatively simpler and tractable computational algorithm even for larger dimensions. In this letter, we will rigorously derive the existence and weak consistency of the maximum pseudo \b{eta}-likelihood estimator in case of MVN mixture models under a reasonable set of assumptions.
Gaussian mixture models are central to classical statistics, widely used in the information sciences, and have a rich mathematical structure. We examine their maximum likelihood estimates through the lens of algebraic statistics. The MLE is not an algebraic function of the data, so there is no notion of ML degree for these models. The critical points of the likelihood function are transcendental, and there is no bound on their number, even for mixtures of two univariate Gaussians.
We derive an asymptotic expansion for the log likelihood of Gaussian mixture models (GMMs) with equal covariance matrices in the low signal-to-noise regime. The expansion reveals an intimate connection between two types of algorithms for parameter estimation: the method of moments and likelihood optimizing algorithms such as Expectation-Maximization (EM). We show that likelihood optimization in the low SNR regime reduces to a sequence of least squares optimization problems that match the moments of the estimate to the ground truth moments one by one. This connection is a stepping stone toward …
Max-mixture processes are defined as Z = max(aX, (1 -- a)Y) with X an asymptotic dependent (AD) process, Y an asymptotic independent (AI) process and a $\in$ [0, 1]. So that, the mixing coefficient a may reveal the strength of the AD part present in the max-mixture process. In this paper we focus on two tests based on censored pairwise likelihood estimates. We compare their performance through an extensive simulation study. Monte Carlo simulation plays a fundamental tool for asymptotic variance calculations. We apply our tests to daily precipitations from the East of Australia. Drawbacks and p…
When we use the normal mixture model, the optimal number of the components describing the data should be determined. Testing homogeneity is good for this purpose; however, to construct its theory is challenging, since the test statistic does not converge to the $χ^{2}$ distribution even asymptotically. The reason for such asymptotic behavior is that the parameter set describing the null hypothesis (N.H.) contains singularities in the space of the alternative hypothesis (A.H.). Recently, a $\it{Bayesian}$ theory for singular models was developed, and it has elucidated various problems of statis…
The finite Gamma mixture model is often used to describe randomness in income data, insurance data, and data from other applications. The popular likelihood approach, however, does not work for this model because the likelihood function is unbounded, and the maximum likelihood estimator is therefore not well defined. There has been much research into ways to ensure the consistent estimation of the mixing distribution, including placing an upper bound on the shape parameter or adding a penalty to the log-likelihood function. In this paper, we show that if the shape parameter in the finite Gamma…
A weighted likelihood approach for robust fitting of a mixture of multivariate Gaussian components is developed in this work. Two approaches have been proposed that are driven by a suitable modification of the standard EM and CEM algorithms, respectively. In both techniques, the M-step is enhanced by the computation of weights aimed at downweighting outliers. The weights are based on Pearson residuals stemming from robust Mahalanobis-type distances. Formal rules for robust clustering and outlier detection can be also defined based on the fitted mixture model. The behavior of the proposed metho…
We consider a semiparametric mixture of two univariate density functions where one of them is known while the weight and the other function are unknown. Such mixtures have a history of application to the problem of detecting differentially expressed genes under two or more conditions in microarray data. Until now, some additional knowledge about the unknown component (e.g. the fact that it belongs to a location family) has been assumed. As opposed to this approach, we do not assume any additional structure on the unknown density function. For this mixture model, we derive a new sufficient iden…
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…
IBM Research - Haifa（IBM海法研究院）；The Hebrew University of Jerusalem（耶路撒冷希伯来大学）
The likelihood function of a finite mixture model is a non-convex function with multiple local maxima and commonly used iterative algorithms such as EM will converge to different solutions depending on initial conditions. In this paper we ask: is it possible to assess how far we are from the global maximum of the likelihood? Since the likelihood of a finite mixture model can grow unboundedly by centering a Gaussian on a single datapoint and shrinking the covariance, we constrain the problem by assuming that the parameters of the individual models are members of a large discrete set (e.g. estim…
The integrated completed likelihood (ICL) criterion has proven to be a very popular approach in model-based clustering through automatically choosing the number of clusters in a mixture model. This approach effectively maximises the complete data likelihood, thereby including the allocation of observations to clusters in the model selection criterion. However for practical implementation one needs to introduce an approximation in order to estimate the ICL. Our contribution here is to illustrate that through the use of conjugate priors one can derive an exact expression for ICL and so avoiding …
Mixture models are a popular tool in model-based clustering. Such a model is often fitted by a procedure that maximizes the likelihood, such as the EM algorithm. At convergence, the maximum likelihood parameter estimates are typically reported, but in most cases little emphasis is placed on the variability associated with these estimates. In part this may be due to the fact that standard errors are not directly calculated in the model-fitting algorithm, either because they are not required to fit the model, or because they are difficult to compute. The examination of standard errors in model-b…
Efficiently Computed Marginal Likelihoods for Multivariate Mixture Models
Laboratoire de Mathématiques Blaise Pascal, CNRS, Université Clermont Auvergne（布莱斯·帕斯卡数学实验室，法国国家科学研究中心，克莱蒙奥弗涅大学）；CNRS, MAP5, Université Paris Cité（法国国家科学研究中心，MAP5，巴黎西岱大学）；Departments of Statistics and Sociology, University of Washington（华盛顿大学统计与社会学系）
Finite mixtures of multivariate normal distributions have been widely used in empirical applications in diverse fields such as statistical genetics and statistical finance. Testing the number of components in multivariate normal mixture models is a long-standing challenge even in the most important case of testing homogeneity. This paper develops likelihood-based tests of the null hypothesis of $M_0$ components against the alternative hypothesis of $M_0 + 1$ components for a general $M_0 \geq 1$. For heteroscedastic normal mixtures, we propose an EM test and derive the asymptotic distribution …
Approximating the marginal likelihood in mixture models
在Chib（1995）中，提出了一种在贝叶斯设置中近似边缘密度的方法，其中一个突出的应用是估计正常混合物中组分的数量。如Neal（1999）和Fruhabeth-Schnatter（2004）中所指出的，由于众所周知的标签切换问题，近似通常不能提供对真实边际密度的适当近似（Celeux等人，2000年）。虽然有其他替代方案来推导近似的边际密度，但我们重新考虑了这里的原始提议，并在Berkhof et al.（2003）和Lee et al.（2008）中表明，一旦解决了标签转换问题，它就真正近似了边际密度。
In this article, we discuss two specific classes of models - Gaussian Mixture Copula models and Mixture of Factor Analyzers - and the advantages of doing inference with gradient descent using automatic differentiation. Gaussian mixture models are a popular class of clustering methods, that offers a principled statistical approach to clustering. However, the underlying assumption, that every mixing component is normally distributed, can often be too rigid for several real life datasets. In order to to relax the assumption about the normality of mixing components, a new class of parametric mixtu…
In this article, we present the maximum weighted likelihood estimator (MWLE) for robust estimations of heavy-tail finite mixture models (FMM). This is motivated by the complex distributional phenomena of insurance claim severity data, where flexible density estimation tools such as FMM are needed but MLE often produces unstable tail estimates under FMM. Under some regularity conditions, MWLE is proved to be consistent and asymptotically normal. We further prove that the tail index obtained by MWLE is consistent even if the model is misspecified, justifying the robustness of MWLE in estimating …
University of California, Berkeley（加州大学伯克利分校）；Carnegie Mellon University（卡内基梅隆大学）
We provide two fundamental results on the population (infinite-sample) likelihood function of Gaussian mixture models with $M \geq 3$ components. Our first main result shows that the population likelihood function has bad local maxima even in the special case of equally-weighted mixtures of well-separated and spherical Gaussians. We prove that the log-likelihood value of these bad local maxima can be arbitrarily worse than that of any global optimum, thereby resolving an open question of Srebro (2007). Our second main result shows that the EM algorithm (or a first-order variant of it) with ran…
There has been much recent interest in modifying Bayesian inference for misspecified models so that it is useful for specific purposes. One popular modified Bayesian inference method is "cutting feedback" which can be used when the model consists of a number of coupled modules, with only some of the modules being misspecified. Cutting feedback methods represent the full posterior distribution in terms of conditional and sequential components, and then modify some terms in such a representation based on the modular structure for specification or computation of a modified posterior distribution.…
Statistical models of unobserved heterogeneity are typically formalized as mixtures of simple parametric models and interest naturally focuses on testing for homogeneity versus general mixture alternatives. Many tests of this type can be interpreted as $C(α)$ tests, as in Neyman (1959), and shown to be locally, asymptotically optimal. These $C(α)$ tests will be contrasted with a new approach to likelihood ratio testing for general mixture models. The latter tests are based on estimation of general nonparametric mixing distribution with the Kiefer and Wolfowitz (1956) maximum likelihood estimat…
Nonparametric maximum likelihood (NPML) for mixture models is a technique for estimating mixing distributions that has a long and rich history in statistics going back to the 1950s, and is closely related to empirical Bayes methods. Historically, NPML-based methods have been considered to be relatively impractical because of computational and theoretical obstacles. However, recent work focusing on approximate NPML methods suggests that these methods may have great promise for a variety of modern applications. Building on this recent work, a class of flexible, scalable, and easy to implement ap…
In this manuscript, we consider a finite multivariate nonparametric mixture model where the dependence between the marginal densities is modeled using the copula device. Pseudo EM stochastic algorithms were recently proposed to estimate all of the components of this model under a location-scale constraint on the marginals. Here, we introduce a deterministic algorithm that seeks to maximize a smoothed semiparametric likelihood. No location-scale assumption is made about the marginals. The algorithm is monotonic in one special case, and, in another, leads to ``approximate monotonicity'' -- where…
Bayesian inference for Continuous-Time Markov Chains (CTMCs) on countably infinite spaces is notoriously difficult because evaluating the likelihood exactly is intractable. One way to address this challenge is to first build a non-negative and unbiased estimate of the likelihood -- involving the matrix exponential of finite truncations of the true rate matrix -- and then to use the estimates in a pseudo-marginal inference method. In this work, we show that we can dramatically increase the efficiency of this approach by avoiding the computation of exact matrix exponentials. In particular, we de…
Mixture models of Plackett-Luce (PL) -- one of the most fundamental ranking models -- are an active research area of both theoretical and practical significance. Most previously proposed parameter estimation algorithms instantiate the EM algorithm, often with random initialization. However, such an initialization scheme may not yield a good initial estimate and the algorithms require multiple restarts, incurring a large time complexity. As for the EM procedure, while the E-step can be performed efficiently, maximizing the log-likelihood in the M-step is difficult due to the combinatorial natur…
University of Sheffield（谢菲尔德大学）；INRIA-ENS（法国国家信息与自动化研究所-巴黎高等师范学校）；Indian Institute of Science（印度科学理工学院）
Several machine learning problems arising in natural language processing can be modeled as a sequence labeling problem. We provide Gaussian process models based on pseudo-likelihood approximation to perform sequence labeling. Gaussian processes (GPs) provide a Bayesian approach to learning in a kernel based framework. The pseudo-likelihood model enables one to capture long range dependencies among the output components of the sequence without becoming computationally intractable. We use an efficient variational Gaussian approximation method to perform inference in the proposed model. We also p…
We propose two neural network based mixture models in this article. The proposed mixture models are explicit in nature. The explicit models have analytical forms with the advantages of computing likelihood and efficiency of generating samples. Computation of likelihood is an important aspect of our models. Expectation-maximization based algorithms are developed for learning parameters of the proposed models. We provide sufficient conditions to realize the expectation-maximization based learning. The main requirements are invertibility of neural networks that are used as generators and Jacobian…
Recent work has shown that deep generative models assign higher likelihood to out-of-distribution inputs than to training data. We show that a factor underlying this phenomenon is a mismatch between the nature of the prior distribution and that of the data distribution, a problem found in widely used deep generative models such as VAEs and Glow. While a typical choice for a prior distribution is a standard Gaussian distribution, properties of distributions of real data sets may not be consistent with a unimodal prior distribution. This paper focuses on the relationship between the choice of a …
StepMix: A Python Package for Pseudo-Likelihood Estimation of Generalized Mixture Models with External Variables
Université de Montréal（蒙特利尔大学）；Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；Massachusetts Institute of Technology（麻省理工学院）；Leiden University（莱顿大学）；Institut universitaire en santé mentale de Montréal（蒙特利尔大学精神卫生大学研究所）
本文介绍开源 Python 包 StepMix，它在统一 EM 框架下实现带协变量与远端结局的广义混合模型的一步、两步及偏差校正三步伪似然估计，并兼容 scikit-learn 与 R。
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…
University of Helsinki（赫尔辛基大学）；Åbo Akademi University（奥博学术大学）
We propose a Bayesian approximate inference method for learning the dependence structure of a Gaussian graphical model. Using pseudo-likelihood, we derive an analytical expression to approximate the marginal likelihood for an arbitrary graph structure without invoking any assumptions about decomposability. The majority of the existing methods for learning Gaussian graphical models are either restricted to decomposable graphs or require specification of a tuning parameter that may have a substantial impact on learned structures. By combining a simple sparsity inducing prior for the graph struct…
The von Mises-Fisher distribution is one of the most widely used probability distributions to describe directional data. Finite mixtures of von Mises-Fisher distributions have found numerous applications. However, the likelihood function for the finite mixture of von Mises-Fisher distributions is unbounded and consequently the maximum likelihood estimation is not well defined. To address the problem of likelihood degeneracy, we consider a penalized maximum likelihood approach whereby a penalty function is incorporated. We prove strong consistency of the resulting estimator. An Expectation-Maxi…
Skew normal mixture models provide a more flexible framework than the popular normal mixtures for modelling heterogeneous data with asymmetric behaviors. Due to the unboundedness of likelihood function and the divergency of shape parameters, the maximum likelihood estimators of the parameters of interest are often not well defined, leading to dissatisfactory inferential process. We put forward a proposal to deal with these issues simultaneously in the context of penalizing the likelihood function. The resulting penalized maximum likelihood estimator is proved to be strongly consistent when the…
Motivated by studies in biological sciences to detect differentially expressed genes, a semiparametric two-component mixture model with one known component is being studied in this paper. Assuming the density of the unknown component to be log-concave, which contains a very broad family of densities, we develop a semiparametric maximum likelihood estimator and propose an EM algorithm to compute it. Our new estimation method finds the mixing proportions and the distribution of the unknown component simultaneously. We establish the identifiability of the proposed semiparametric mixture model and…
Testing for regime switching when the regime switching probabilities are specified either as constants (`mixture models') or are governed by a finite-state Markov chain (`Markov switching models') are long-standing problems that have also attracted recent interest. This paper considers testing for regime switching when the regime switching probabilities are time-varying and depend on observed data (`observation-dependent regime switching'). Specifically, we consider the likelihood ratio test for observation-dependent regime switching in mixture autoregressive models. The testing problem is hig…
