Related papers: A Probabilistic Algorithm for Computing Data-Discr…
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 develop a probabilistic algorithm for computing elimination ideals of likelihood equations, which is for larger models by far more efficient than directly computing Groebner bases or the interpolation method proposed in the first…
In exploratory factor analysis, model parameters are usually estimated by maximum likelihood method. The maximum likelihood estimate is obtained by solving a complicated multivariate algebraic equation. Since the solution to the equation is…
We present a numerical algorithm for finding real non-negative solutions to polynomial equations. Our methods are based on the expectation maximization and iterative proportional fitting algorithms, which are used in statistics to find…
Let $f, f_1, \ldots, f_\nV$ be polynomials with rational coefficients in the indeterminates $\bfX=X_1, \ldots, X_n$ of maximum degree $D$ and $V$ be the set of common complex solutions of $\F=(f_1,\ldots, f_\nV)$. We give an algorithm…
Let $\RR$ be a real closed field (e.g. the field of real numbers) and $\mathscr{S} \subset \RR^n$ be a semi-algebraic set defined as the set of points in $\RR^n$ satisfying a system of $s$ equalities and inequalities of multivariate…
In many fields of science, generalized likelihood ratio tests are established tools for statistical inference. At the same time, it has become increasingly common that a simulator (or generative model) is used to describe complex processes…
We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of…
Given a model in algebraic statistics and some data, the likelihood function is a rational function on a projective variety. Algebraic algorithms are presented for computing all critical points of this function, with the aim of identifying…
Gradient-based solvers risk convergence to local optima, leading to incorrect researcher inference. Heuristic-based algorithms are able to ``break free" of these local optima to eventually converge to the true global optimum. However, given…
Estimating the parameters of ordinary differential equations (ODEs) is of fundamental importance in many scientific applications. While ODEs are typically approximated with deterministic algorithms, new research on probabilistic solvers…
Maximum likelihood estimation is a fundamental optimization problem in statistics. We study this problem on manifolds of matrices with bounded rank. These represent mixtures of distributions of two independent discrete random variables. We…
Estimating a Gibbs density function given a sample is an important problem in computational statistics and statistical learning. Although the well established maximum likelihood method is commonly used, it requires the computation of the…
We consider systems of polynomial equations and inequalities in $\mathbb{Q}[\boldsymbol{y}][\boldsymbol{x}]$ where $\boldsymbol{x} = (x_1, \ldots, x_n)$ and $\boldsymbol{y} = (y_1, \ldots,y_t)$. The $\boldsymbol{y}$ indeterminates are…
Many probabilistic inference tasks involve summations over exponentially large sets. Recently, it has been shown that these problems can be reduced to solving a polynomial number of MAP inference queries for a model augmented with randomly…
Linear and Quadratic Discriminant Analysis are well-known classical methods but can heavily suffer from non-Gaussian distributions and/or contaminated datasets, mainly because of the underlying Gaussian assumption that is not robust. To…
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…
Most statistical software packages implement numerical strategies for computation of maximum likelihood estimates in random effects models. Little is known, however, about the algebraic complexity of this problem. For the one-way layout…
Feature attribution methods, or saliency maps, are one of the most popular approaches for explaining the decisions of complex machine learning models such as deep neural networks. In this study, we propose a stochastic optimization approach…
Gaussian Process (GP) models are popular statistical surrogates used for emulating computationally expensive computer simulators. The quality of a GP model fit can be assessed by a goodness of fit measure based on optimized likelihood.…