Contributions to computational algebraic statistics.
The field of algebraic statistics arose from the realization that problems in statistics can often be viewed, and subsequently solved, using algebraic concepts and tools. In this work, we explore the application of algebraic statistical methods to problems of the analysis of contingency tables. The landmark paper of Diaconis and Sturmfels (1998) introduced an MCMC sampler to aid in conducting exact conditional inference for discrete exponential family models. This sampler is subject to problems of mixing and efficiency, just as any MCMC sampler is. In chapter one, we introduce practical acceleration schemes that exploit the structure of the problem. We also introduce implementations of these schemes inside the metropolis()function in algstat. The methods introduced by Diaconis and Sturmfels can also be extended to the Poisson and logistic regression settings. These methods have been extensively studied but no general-purpose implementation exists. In chapter two, we introduce aglm(), a function that fits algebraic Poisson and logistic regression models. In chapter three, we explore an algebraic approach to minimum chi-square estimation. Minimum chi-square estimation, a statistical estimation paradigm alternative to maximum likelihood estimation, exhibits a strong algebraic structure. The computation of a minimum chi-square estimator involves an optimization problem with rich algebraic structure: a rational objective function over an algebraic feasibility region. This structure can be exploited with Bertini, a state-of-the-art software package from the algebraic geometry community. We introduce enhancements to bertini, an R package that connects to Bertini, that enables the user to compute minimum chi-square estimators.