Maximum likelihood and polynomial system solving

The use of maximum likelihood estimation has become extremely popular in a vast number of fields. Statistical methods are paramount for analysing biological data and maximum likelihood estimation therefore is the dominant framework in the different fields of computational biology. Hidden Markov Models, for example, are statistical models for discrete data in which the system being modeled is assumed to be a Markov process with unobserved states. They can be considered as the simplest dynamic Bayesian networks and were first used for speech recognition in the mid-1970´s. In the second half of the 1980´s they were first used for modeling biological sequences and have since then become ubiquitous in the field of bioinformatics. Some of their applications are in DNA sequence alignment, gene finding, phylogenetics and much more. The most common method for finding maximum likelihood estimates is Expectation Maximization (EM). This paper seeks out to establish a numerical method for finding maximum likelihood estimates which is guaranteed to find the global maximum. This is achieved by first showing that for algebraic statistical models maximum likelihood estimation of the model parameters corresponds with solving a polynomial system. Then an algorithm is presented which allows to find all solutions of polynomial systems by solving a generalized eigenvalue problem.