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 not only derive - for the first time - Bernstein-like large deviation bounds for the missing mass whose exponents behave almost linearly with respect to deviation size, but also sharpen McAllester and Ortiz (2003) and Berend and Kontorovich (2013) for large sample sizes in the case of small deviations which is the most interesting case in learning theory. In the meantime, our approach shows that the heterogeneity issue introduced in McAllester and Ortiz (2003) is resolvable in the case of missing mass in the sense that one can use standard inequalities but it may not lead to strong results. Thus, we postulate that our results are general and can be applied to provide potentially sharp Bernstein-like bounds under some constraints.
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…
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…
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…
In this paper, a scale mixture of Normal distributions model is developed for classification and clustering of data having outliers and missing values. The classification method, based on a mixture model, focuses on the introduction of latent variables that gives us the possibility to handle sensitivity of model to outliers and to allow a less restrictive modelling of missing data. Inference is processed through a Variational Bayesian Approximation and a Bayesian treatment is adopted for model learning, supervised classification and clustering.
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…
In data-mining applications, we are frequently faced with a large fraction of missing entries in the data matrix, which is problematic for most discriminant machine learning algorithms. A solution that we explore in this paper is the use of a generative model (a mixture of Gaussians) to compute the conditional expectation of the missing variables given the observed variables. Since training a Gaussian mixture with many different patterns of missing values can be computationally very expensive, we introduce a spanning-tree based algorithm that significantly speeds up training in these condition…
University of Oxford（牛津大学）；CEREMADE, CNRS Université Paris-Dauphine（巴黎多芬大学CNRS-CEREMADE实验室）
We propose a variational autoencoder architecture to model both ignorable and nonignorable missing data using pattern-set mixtures as proposed by Little (1993). Our model explicitly learns to cluster the missing data into missingness pattern sets based on the observed data and missingness masks. Underpinning our approach is the assumption that the data distribution under missingness is probabilistically semi-supervised by samples from the observed data distribution. Our setup trades off the characteristics of ignorable and nonignorable missingness and can thus be applied to data of both types.…
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…
We consider the situation where the observed sample contains some observations whose class of origin is known (that is, they are classified with respect to the g underlying classes of interest), and where the remaining observations in the sample are unclassified (that is, their class labels are unknown). For class-conditional distributions taken to be known up to a vector of unknown parameters, the aim is to estimate the Bayes' rule of allocation for the allocation of subsequent unclassified observations. Estimation on the basis of both the classified and unclassified data can be undertaken in…
ASELSAN Research Center（ASELSAN研究中心）；Bilkent University（比尔肯特大学）；DataBoss A.S.（DataBoss公司）
We investigate regression for variable length sequential data containing missing samples and introduce a novel tree architecture based on the Long Short-Term Memory (LSTM) networks. In our architecture, we employ a variable number of LSTM networks, which use only the existing inputs in the sequence, in a tree-like architecture without any statistical assumptions or imputations on the missing data, unlike all the previous approaches. In particular, we incorporate the missingness information by selecting a subset of these LSTM networks based on "presence-pattern" of a certain number of previous …
We give an algorithm for completing an order-$m$ symmetric low-rank tensor from its multilinear entries in time roughly proportional to the number of tensor entries. We apply our tensor completion algorithm to the problem of learning mixtures of product distributions over the hypercube, obtaining new algorithmic results. If the centers of the product distribution are linearly independent, then we recover distributions with as many as $Ω(n)$ centers in polynomial time and sample complexity. In the general case, we recover distributions with as many as $\tildeΩ(n)$ centers in quasi-polynomial ti…
Brown University（布朗大学）；IAS（高等研究院）；Purdue University（普渡大学）
Given a mixture between two populations of coins, "positive" coins that each have -- unknown and potentially different -- bias $\geq\frac{1}{2}+Δ$ and "negative" coins with bias $\leq\frac{1}{2}-Δ$, we consider the task of estimating the fraction $ρ$ of positive coins to within additive error $ε$. We achieve an upper and lower bound of $Θ(\fracρ{ε^2Δ^2}\log\frac{1}δ)$ samples for a $1-δ$ probability of success, where crucially, our lower bound applies to all fully-adaptive algorithms. Thus, our sample complexity bounds have tight dependence for every relevant problem parameter. A crucial compo…
