Solving the Likelihood Equations

Abstract

Given a model in algebraic statistics and 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 the local maxima in the probability simplex. Applications include models specified by rank conditions on matrices and the Jukes-Cantor models of phylogenetics. The maximum likelihood degree of a generic complete intersection is also determined.

Lectures on Algebraic Statistics

“…It is tacitly assumed that s < k. If P has k or more generators then we can obtain a bound by replacing P by a suitable subideal. The following result appeared as Theorem 5 in [61]. …”

“…We refer to [61] for details on the practical implementation of the algorithm, including the delicate work of computing in the quotient ring R[V ] = R[p 1 . .…”

“…The saturation in Step 4 requires some tricks in order to run efficiently. In the numerical experiments in [61], for many models and most data, it sufficed to saturate only once with respect to a single polynomial, as follows:…”

Experiments with the Site Frequency Spectrum

“…Exact maximum likelihood estimation (e.g., Yang 2000; Hosten et al 2005;Casanellas et al 2005) as well as exact posterior sampling (Sainudiin and York 2009) is only feasible for small sample sizes (n ≤ 4). The standard approach is to rely on Monte Carlo Markov chain (MCMC) algorithms (Metropolis et al 1953;Hastings 1970) to obtain dependent samples from the posterior under the assumption that the algorithm has converged to the desired stationary distribution.…”

“…In an ideal world, the optimal inference procedure would be based on the minimally sufficient statistic and implemented in a computing environment free of engineering constraints. Unfortunately, minimally sufficient statistics of data at the currently finest resolution of U m n are unknown beyond the simplest models of mutation with small values of n (Yang 2000;Hosten et al 2005;Casanellas et al 2005;Sainudiin and York 2009). Computationally-intensive inference, based on an observed u o ∈ U m n , with realistically large n and m, is currently impossible for recombining loci and prohibitive for non-recombining loci.…”