Related papers: Bounding the maximum likelihood degree
We present a streamlined proof of the foundational result in the theory of exponential random graph models (ERGMs) that the maximum likelihood estimate exists if and only if the target statistic lies in the relative interior of the convex…
A hidden Markov model with trends is a hidden Markov model whose emission distributions are translated by a trend that depends on the current hidden state and on the current time. Contrary to standard hidden Markov models, such processes…
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…
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…
For a given parametric probability model, we consider the risk of the maximum likelihood estimator with respect to $\alpha$-divergence, which includes the special cases of Kullback--Leibler divergence, the Hellinger distance and $\chi^2$…
We consider upper exponential bounds for the probability of the event that an absolute deviation of sample mean from mathematical expectation p is bigger comparing with some ordered level epsilon. These bounds include 2 coefficients {alpha,…
We study multivariate Gaussian models that are described by linear conditions on the concentration matrix. We compute the maximum likelihood (ML) degrees of these models. That is, we count the critical points of the likelihood function over…
As is the case for many curved exponential families, the computation of maximum likelihood estimates in a multivariate normal model with a Kronecker covariance structure is typically carried out with an iterative algorithm, specifically, a…
For general data, the number of complex solutions to the likelihood equations is constant and this number is called the (maximum likelihood) ML-degree of the model. In this article, we describe the special locus of data for which the…
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…
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…
Maximum likelihood estimation is effective for identifying dynamical systems, but applying it to large networks becomes computationally prohibitive. This paper introduces a maximum likelihood estimation method that enables identification of…
The manifold hypothesis says that natural high-dimensional data lie on or around a low-dimensional manifold. The recent success of statistical and learning-based methods in very high dimensions empirically supports this hypothesis,…
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…
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…
Maximum likelihood estimation (MLE) is a fundamental computational problem in statistics. The problem is to maximize the likelihood function with respect to given data on a statistical model. An algebraic approach to this problem is to…
We import the algebro-geometric notion of a complete collineation into the study of maximum likelihood estimation in directed Gaussian graphical models. A complete collineation produces a perturbation of sample data, which we call a…
In this paper, we study sample size thresholds for maximum likelihood estimation for tensor normal models. Given the model parameters and the number of samples, we determine whether, almost surely, (1) the likelihood function is bounded…
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…
We propose and study properties of maximum likelihood estimators in the class of conditional transformation models. Based on a suitable explicit parameterisation of the unconditional or conditional transformation function, we establish a…