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We study the problem of maximum likelihood estimation of densities that are log-concave and lie in the graphical model corresponding to a given undirected graph $G$. We show that the maximum likelihood estimate (MLE) is the product of the…
We consider distributed estimation of the inverse covariance matrix, also called the concentration or precision matrix, in Gaussian graphical models. Traditional centralized estimation often requires global inference of the covariance…
We study maximum likelihood estimation in log-linear models under conditional Poisson sampling schemes. We derive necessary and sufficient conditions for existence of the maximum likelihood estimator (MLE) of the model parameters and…
We study maximum likelihood estimation in Gaussian graphical models from a geometric point of view. An algebraic elimination criterion allows us to find exact lower bounds on the number of observations needed to ensure that the maximum…
The existence of the maximum likelihood estimate in hierarchical loglinear models is crucial to the reliability of inference for this model. Determining whether the estimate exists is equivalent to finding whether the sufficient statistics…
Graphical models with bi-directed edges (<->) represent marginal independence: the absence of an edge between two vertices indicates that the corresponding variables are marginally independent. In this paper, we consider maximum likelihood…
A discrete statistical model is a subset of a probability simplex. Its maximum likelihood estimator (MLE) is a retraction from that simplex onto the model. We characterize all models for which this retraction is a rational function. This is…
We solve the problem of estimating the distribution of presumed i.i.d. observations for the total variation loss. Our approach is based on density models and is versatile enough to cope with many different ones, including some density…
We consider discrete graphical models Markov with respect to a graph $G$ and propose two distributed marginal methods to estimate the maximum likelihood estimate of the canonical parameter of the model. Both methods are based on a…
Maximum likelihood (ML) estimation is widely used in statistics. The h-likelihood has been proposed as an extension of Fisher's likelihood to statistical models including unobserved latent variables of recent interest. Its advantage is that…
We study maximum likelihood estimation for the statistical model for undirected random graphs, known as the $\beta$-model, in which the degree sequences are minimal sufficient statistics. We derive necessary and sufficient conditions, based…
We provide a polyhedral description of the conditions for the existence of the maximum likelihood estimate (MLE) for a hierarchical log-linear model. The MLE exists if and only if the observed margins lie in the relative interior of the…
This paper considers the maximum likelihood estimation of factor models of high dimension, where the number of variables (N) is comparable with or even greater than the number of observations (T). An inferential theory is developed. We…
We explore the problems of classification of composite object (images, speech signals) with low number of models per class. We study the question of improving recognition performance for medium-sized database (thousands of classes). The key…
In a regular full exponential family, the maximum likelihood estimator (MLE) need not exist in the traditional sense. However, the MLE may exist in the completion of the exponential family. Existing algorithms for finding the MLE in the…
Given a statistical model, the maximum likelihood degree is the number of complex solutions to the likelihood equations for generic data. We consider discrete algebraic statistical models and study the solutions to the likelihood equations…
Maximum likelihood estimation (MLE) is a fundamental problem in statistics. Characteristics of the MLE problem for discrete algebraic statistical models are reflected in the geometry of the $\textit{likelihood correspondence}$, a variety…
We propose an efficient algorithm for approximate computation of the profile maximum likelihood (PML), a variant of maximum likelihood maximizing the probability of observing a sufficient statistic rather than the empirical sample. The PML…
In order to learn the complex features of large spatio-temporal data, models with large parameter sets are often required. However, estimating a large number of parameters is often infeasible due to the computational and memory costs of…
The assumption of log-concavity is a flexible and appealing nonparametric shape constraint in distribution modelling. In this work, we study the log-concave maximum likelihood estimator (MLE) of a probability mass function (pmf). We show…