We determine an explicit Gr ?obner basis, consisting of linear forms and determinantal quadrics, for the prime ideal of Raftery’s mixture... Show moreWe 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. Show less
We study the critical points of the likelihood function over the Fermat hypersurface. This problem is related to one of the main problems in... Show moreWe 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. Show less
