Finite mixture models are statistical models which appear in many problems in statistics and machine learning. In such models it is assumed that data are drawn from random probability measures, called mixture components, which are themselves drawn from a probability measure P over probability measures. When estimating mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this paper, we make no assumption on the mixture components, and instead assume that observations from the mixture model are grouped, such that observations in the same group are known to be drawn from the same component. We show that any mixture of m probability measures can be uniquely identified provided there are 2m-1 observations per group. Moreover we show that, for any m, there exists a mixture of m probability measures that cannot be uniquely identified when groups have 2m-2 observations. Our results hold for any sample space with more than one element.
Technische Universität Berlin（柏林工业大学）；Berlin Institute for the Foundations of Learning and Data（柏林学习与数据基础研究所）
Recent work has shown that finite mixture models with $m$ components are identifiable, while making no assumptions on the mixture components, so long as one has access to groups of samples of size $2m-1$ which are known to come from the same mixture component. In this work we generalize that result and show that, if every subset of $k$ mixture components of a mixture model are linearly independent, then that mixture model is identifiable with only $(2m-1)/(k-1)$ samples per group. We further show that this value cannot be improved. We prove an analogous result for a stronger form of identifiab…
University of Michigan（密歇根大学）；Technische Universität Berlin（柏林工业大学）
Recent research has established sufficient conditions for finite mixture models to be identifiable from grouped observations. These conditions allow the mixture components to be nonparametric and have substantial (or even total) overlap. This work proposes an algorithm that consistently estimates any identifiable mixture model from grouped observations. Our analysis leverages an oracle inequality for weighted kernel density estimators of the distribution on groups, together with a general result showing that consistent estimation of the distribution on groups implies consistent estimation of m…
Motivated by problems in data clustering, we establish general conditions under which families of nonparametric mixture models are identifiable, by introducing a novel framework involving clustering overfitted \emph{parametric} (i.e. misspecified) mixture models. These identifiability conditions generalize existing conditions in the literature, and are flexible enough to include for example mixtures of Gaussian mixtures. In contrast to the recent literature on estimating nonparametric mixtures, we allow for general nonparametric mixture components, and instead impose regularity assumptions on …
The problem of identifiability of finite mixtures of finite product measures is studied. A mixture model with $K$ mixture components and $L$ observed variables is considered, where each variable takes its value in a finite set with cardinality $M$.The variables are independent in each mixture component. The identifiability of a mixture model means the possibility of attaining the mixture components parameters by observing its mixture distribution. In this paper, we investigate fundamental relations between the identifiability of mixture models and the separability of their observed variables b…
University of Minnesota（明尼苏达大学）；Oregon State University（俄勒冈州立大学）；University of Virginia（弗吉尼亚大学）；University of Florida（佛罗里达大学）
Linear mixture models have proven very useful in a plethora of applications, e.g., topic modeling, clustering, and source separation. As a critical aspect of the linear mixture models, identifiability of the model parameters is well-studied, under frameworks such as independent component analysis and constrained matrix factorization. Nevertheless, when the linear mixtures are distorted by an unknown nonlinear functions -- which is well-motivated and more realistic in many cases -- the identifiability issues are much less studied. This work proposes an identification criterion for a nonlinear m…
This paper studies identifiability and convergence behaviors for parameters of multiple types in finite mixtures, and the effects of model fitting with extra mixing components. First, we present a general theory for strong identifiability, which extends from the previous work of Nguyen [2013] and Chen [1995] to address a broad range of mixture models and to handle matrix-variate parameters. These models are shown to share the same Wasserstein distance based optimal rates of convergence for the space of mixing distributions --- $n^{-1/2}$ under $W_1$ for the exact-fitted and $n^{-1/4}$ under $W…
University of Wisconsin-Madison（威斯康星大学麦迪逊分校）；Universität Duisburg-Essen（杜伊斯堡-埃森大学）；University of Cambridge（剑桥大学）
Mixtures of ranking models are standard tools for ranking problems. However, even the fundamental question of parameter identifiability is not fully understood: the identifiability of a mixture model with two Bradley-Terry-Luce (BTL) components has remained open. In this work, we show that popular mixtures of ranking models with two components (BTL, multinomial logistic models with slates of size 3, or Plackett-Luce) are generically identifiable, i.e., the ground-truth parameters can be identified except when they are from a pathological subset of measure zero. We provide a framework for verif…
University of Minnesota（明尼苏达大学）；University of Virginia（弗吉尼亚大学）
We study the problem of learning a mixture model of non-parametric product distributions. The problem of learning a mixture model is that of finding the component distributions along with the mixing weights using observed samples generated from the mixture. The problem is well-studied in the parametric setting, i.e., when the component distributions are members of a parametric family -- such as Gaussian distributions. In this work, we focus on multivariate mixtures of non-parametric product distributions and propose a two-stage approach which recovers the component distributions of the mixture…
Missing data problems arise in many applied research studies. They may jeopardize statistical inference of the model of interest, if the missing mechanism is nonignorable, that is, the missing mechanism depends on the missing values themselves even conditional on the observed data. With a nonignorable missing mechanism, the model of interest is often not identifiable without imposing further assumptions. We find that even if the missing mechanism has a known parametric form, the model is not identifiable without specifying a parametric outcome distribution. Although it is fundamental for valid…
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…
Synthetic Potential Outcomes and Causal Mixture Identifiability
Eric and Wendy Schmidt Center（埃里克和温迪·施密特中心）；Broad Institute of MIT and Harvard（麻省理工学院与哈佛大学布罗德研究所）；Laboratory for Information and Decision Systems（信息与决策系统实验室）；MIT-IBM Watson AI Lab（麻省理工学院-IBM沃森人工智能实验室）
Mixed Membership Models (MMMs) are a popular family of latent structure models for complex multivariate data. Instead of forcing each subject to belong to a single cluster, MMMs incorporate a vector of subject-specific weights characterizing partial membership across clusters. With this flexibility come challenges in uniquely identifying, estimating, and interpreting the parameters. In this article, we propose a new class of Dimension-Grouped MMMs (Gro-M$^3$s) for multivariate categorical data, which improve parsimony and interpretability. In Gro-M$^3$s, observed variables are partitioned into…
When estimating finite mixture models, it is common to make assumptions on the mixture components, such as parametric assumptions. In this work, we make no distributional assumptions on the mixture components and instead assume that observations from the mixture model are grouped, such that observations in the same group are known to be drawn from the same mixture component. We precisely characterize the number of observations $n$ per group needed for the mixture model to be identifiable, as a function of the number $m$ of mixture components. In addition to our assumption-free analysis, we als…
University of Chicago（芝加哥大学）；Carnegie Mellon University（卡内基梅隆大学）
We prove identifiability of a broad class of deep latent variable models that (a) have universal approximation capabilities and (b) are the decoders of variational autoencoders that are commonly used in practice. Unlike existing work, our analysis does not require weak supervision, auxiliary information, or conditioning in the latent space. Specifically, we show that for a broad class of generative (i.e. unsupervised) models with universal approximation capabilities, the side information $u$ is not necessary: We prove identifiability of the entire generative model where we do not observe $u$ a…
False discovery rate (FDR) control in structured hypotheses testing is an important topic in simultaneous inference. Most existing methods that aim to utilize group structure among hypotheses either employ the groupwise mixture model or weight all p-values or hypotheses. Thus, their powers can be improved when the groupwise mixture model is inappropriate or when most groups contain only true null hypotheses. Motivated by this, we propose a grouped, selectively weighted FDR procedure, which we refer to as "sGBH". Specifically, without employing the groupwise mixture model, sGBH identifies group…
In this paper, we consider mixtures of multinomial logistic models (MNL), which are known to $ε$-approximate any random utility model. Despite its long history and broad use, rigorous results are only available for learning a uniform mixture of two MNLs. Continuing this line of research, we study the problem of learning an arbitrary mixture of two MNLs. We show that the identifiability of the mixture models may only fail on an algebraic variety of a negligible measure. This is done by reducing the problem of learning a mixture of two MNLs to the problem of solving a system of univariate quarti…
A mixture of joint generalized hyperbolic distributions (MJGHD) is introduced for asymmetric clustering for high-dimensional data. The MJGHD approach takes into account the cluster-specific subspace, thereby limiting the number of parameters to estimate while also facilitating visualization of results. Identifiability is discussed, and a multi-cycle ECM algorithm is outlined for parameter estimation. The MJGHD approach is illustrated on two real data sets, where the Bayesian information criterion is used for model selection.
A well known identifiability issue in factor analytic models is the invariance with respect to orthogonal transformations. This problem burdens the inference under a Bayesian setup, where Markov chain Monte Carlo (MCMC) methods are used to generate samples from the posterior distribution. We introduce a post-processing scheme in order to deal with rotation, sign and permutation invariance of the MCMC sample. The exact version of the contributed algorithm requires to solve $2^q$ assignment problems per (retained) MCMC iteration, where $q$ denotes the number of factors of the fitted model. For l…
We propose the supervised hierarchical Dirichlet process (sHDP), a nonparametric generative model for the joint distribution of a group of observations and a response variable directly associated with that whole group. We compare the sHDP with another leading method for regression on grouped data, the supervised latent Dirichlet allocation (sLDA) model. We evaluate our method on two real-world classification problems and two real-world regression problems. Bayesian nonparametric regression models based on the Dirichlet process, such as the Dirichlet process-generalised linear models (DP-GLM) h…
In many real-world scenarios, such as gene knockout experiments, targeted interventions are often accompanied by unknown interventions at off-target sites. Moreover, different units can get randomly exposed to different unknown interventions, thereby creating a mixture of interventions. Identifying different components of this mixture can be very valuable in some applications. Motivated by such situations, in this work, we study the problem of identifying all components present in a mixture of interventions on a given causal Bayesian Network. We construct an example to show that, in general, t…
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…
We study the problem of separating a mixture of distributions, all of which come from interventions on a known causal bayesian network. Given oracle access to marginals of all distributions resulting from interventions on the network, and estimates of marginals from the mixture distribution, we want to recover the mixing proportions of different mixture components. We show that in the worst case, mixing proportions cannot be identified using marginals only. If exact marginals of the mixture distribution were known, under a simple assumption of excluding a few distributions from the mixture, we…
A mixture of multiple scaled generalized hyperbolic distributions (MMSGHDs) is introduced. Then, a coalesced generalized hyperbolic distribution (CGHD) is developed by joining a generalized hyperbolic distribution with a multiple scaled generalized hyperbolic distribution. After detailing the development of the MMSGHDs, which arises via implementation of a multi-dimensional weight function, the density of the mixture of CGHDs is developed. A parameter estimation scheme is developed using the ever-expanding class of MM algorithms and the Bayesian information criterion is used for model selectio…
Unsupervised mixture learning (UML) aims at identifying linearly or nonlinearly mixed latent components in a blind manner. UML is known to be challenging: Even learning linear mixtures requires highly nontrivial analytical tools, e.g., independent component analysis or nonnegative matrix factorization. In this work, the post-nonlinear (PNL) mixture model -- where unknown element-wise nonlinear functions are imposed onto a linear mixture -- is revisited. The PNL model is widely employed in different fields ranging from brain signal classification, speech separation, remote sensing, to causal di…
Prediction Markets as Bayesian Inverse Problems: Uncertainty Quantification, Identifiability, and Information Gain from Price-Volume Histories under Latent Types
The use of a finite mixture of normal distributions in model-based clustering allows to capture non-Gaussian data clusters. However, identifying the clusters from the normal components is challenging and in general either achieved by imposing constraints on the model or by using post-processing procedures. Within the Bayesian framework we propose a different approach based on sparse finite mixtures to achieve identifiability. We specify a hierarchical prior where the hyperparameters are carefully selected such that they are reflective of the cluster structure aimed at. In addition this prior a…
We are concerned in clustering continuous data sets subject to non-ignorable missingness. We perform clustering with a specific semi-parametric mixture, under the assumption of conditional independence given the component. The mixture model isused for clustering and not for estimating the density of the full variables (observed and unobserved), thus we do not need other assumptions on the component distribution neither to specify the missingness mechanism. Estimation is performed by maximizing an extension of smoothed likelihood allowing missingness. This optimization is achieved by a Majoriza…
Recent advances on overfitting Bayesian mixture models provide a solid and straightforward approach for inferring the underlying number of clusters and model parameters in heterogeneous datasets. The applicability of such a framework in clustering correlated high dimensional data is demonstrated. For this purpose an overfitting mixture of factor analyzers is introduced, assuming that the number of factors is fixed. A Markov chain Monte Carlo (MCMC) sampler combined with a prior parallel tempering scheme is used to estimate the posterior distribution of model parameters. The optimal number of f…
