Geometry of Higher-Order Markov Chains

Geometry of Higher-Order Markov Chains AS2012 Special Volume, part 1: This issue includes a second series of papers from talks, posters and collaborations resulting from and inspired by the Algebraic Statistics in the Alleghenies Conference at Penn State, which took place in July 2012. Sturmfels, Bernd Creator Faculty bernd@mis.mpg.de MATH / Applied Mathematics Illinois Institute of Technology Affiliated department Markov chains Gröbner bases Maximum likelihood 2012 2012 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. Article