Maximum Likelihood for Matrices with Rank Constraints

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 determine the maximum likelihood degree for a range of determinantal varieties, and we apply numerical algebraic geometry to compute all critical points of their likelihood functions. This led to the discovery of maximum likelihood duality between matrices of complementary ranks, a result proved subsequently by Draisma and Rodriguez.Comment: 22 pages, 1 figur

Maximum likelihood geometry in the presence of data zeros

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 when the data contain zeros and are no longer generic. Focusing on sampling and model zeros, we show that, in these cases, the solutions to the likelihood equations are contained in a previously studied variety, the likelihood correspondence. The number of these solutions give a lower bound on the ML degree, and the problem of finding critical points to the likelihood function can be partitioned into smaller and computationally easier problems involving sampling and model zeros. We use this technique to compute a lower bound on the ML degree for $2 \times 2 \times 2 \times 2$ tensors of border rank $\leq 2$ and $3 \times n$ tables of rank $\leq 2$ for $n=11, 12, 13, 14$, the first four values of $n$ for which the ML degree was previously unknown

Data-Discriminants of Likelihood Equations

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 solve a very structured parameterized polynomial system called likelihood equations. For general choices of data, the number of complex solutions to the likelihood equations is finite and called the ML-degree of the model. The only solutions to the likelihood equations that are statistically meaningful are the real/positive solutions. However, the number of real/positive solutions is not characterized by the ML-degree. We use discriminants to classify data according to the number of real/positive solutions of the likelihood equations. We call these discriminants data-discriminants (DD). We develop a probabilistic algorithm for computing DDs. Experimental results show that, for the benchmarks we have tried, the probabilistic algorithm is more efficient than the standard elimination algorithm. Based on the computational results, we discuss the real root classification problem for the 3 by 3 symmetric matrix~model.Comment: 2 table

Geometry of Higher-Order Markov Chains

We determine an explicit Gr\"obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery's mixture transition distribution model for Markov chains. When the states are binary, the corresponding projective variety is a linear space, the model itself consists of two simplices in a cross-polytope, and the likelihood function typically has two local maxima. In the general non-binary case, the model corresponds to a cone over a Segre variety.Comment: 9 page

The maximum likelihood degree of Fermat hypersurfaces

We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in statistical optimization: maximum likelihood estimation. The number of critical points over a projective variety is a topological invariant of the variety and is called maximum likelihood degree. We provide closed formulas for the maximum likelihood degree of any Fermat curve in the projective plane and of Fermat hypersurfaces of degree 2 in any projective space. Algorithmic methods to compute the ML degree of a generic Fermat hypersurface are developed throughout the paper. Such algorithms heavily exploit the symmetries of the varieties we are considering. A computational comparison of the different methods and a list of the maximum likelihood degrees of several Fermat hypersurfaces are available in the last section.Comment: Final version. Accepted for publication on Journal of Algebraic Statistic