Related papers: Maximum likelihood geometry in the presence of dat…
We consider statistical models arising from the common set of solutions to a sparse polynomial system with general coefficients. The maximum likelihood degree counts the number of critical points of the likelihood function restricted to the…
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…
Computing all critical points of a monomial on a very affine variety is a fundamental task in algebraic statistics, particle physics and other fields. The number of critical points is known as the maximum likelihood (ML) degree. When the…
We settle a conjecture by Bik and Marigliano stating that the degree of a one-dimensional discrete model with rational maximum likelihood estimator is bounded above by a linear function in the size of its support, therefore showing that…
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…
The asymptotic normality of the maximum likelihood estimator (MLE) under regularity conditions is a cornerstone of statistical theory. In this paper, we give explicit upper bounds on the distributional distance between the distribution of…
The maximum likelihood threshold of a graph is the smallest number of data points that guarantees that maximum likelihood estimates exist almost surely in the Gaussian graphical model associated to the graph. We show that this graph…
Maximum likelihood degree of a projective variety is the number of critical points of a general likelihood function. In this note, we compute the Maximum likelihood degree of Fermat hypersurfaces. We give a formula of the Maximum likelihood…
Relational models for contingency tables are generalizations of log-linear models, allowing effects associated with arbitrary subsets of cells in a possibly incomplete table, and not necessarily containing the overall effect. In this…
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…
The minimum number of observations such that the maximum likelihood estimator in a Gaussian graphical model exists with probability one is called the maximum likelihood threshold of the underlying graph G. The natural algebraic relaxation…
The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. We give a new characterization of…
The choice of free parameters in network models is subjective, since it depends on what topological properties are being monitored. However, we show that the Maximum Likelihood (ML) principle indicates a unique, statistically rigorous…
The complexity of a maximum likelihood estimation is measured by its maximum likelihood degree ($ML$ degree). In this paper we study the maximum likelihood problem associated to chemical networks composed by one single chemical reaction…
Loss tomography has been studied for more than 10 years and a number of estimators have been proposed. The estimators can be divided into two classes: maximum likelihood and non-maximum likelihood. The maximum likelihood estimators rely on…
We revisit the problem of the existence of the maximum likelihood estimate for multi-class logistic regression. We show that one method of ensuring its existence is by assigning positive probability to every class in the sample dataset. The…
In this paper we revisit the likelihood geometry of Gaussian graphical models. We give a detailed proof that the ML-degree behaves monotonically on induced subgraphs. Furthermore, we complete a missing argument that the ML-degree of the…
This note introduces the $\texttt{LikelihoodGeometry}$ package for the computer algebra system $\textit{Macaulay2}$. This package gives tools to construct the likelihood correspondence of a discrete algebraic statistical model, a variety…
Numerical nonlinear algebra is applied to maximum likelihood estimation for Gaussian models defined by linear constraints on the covariance matrix. We examine the generic case as well as special models (e.g. Toeplitz, sparse, trees) that…
The maximum likelihood threshold (MLT) of a graph $G$ is the minimum number of samples to almost surely guarantee existence of the maximum likelihood estimate in the corresponding Gaussian graphical model. Recently a new characterization of…