Geometry of Higher-Order Markov Chains
We determine an explicit Grobner 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.
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