UC Berkeley

Algebraic statistics is a relatively young field, which explores how algebra and statistics interact, thus fostering a meaningful dialogue between theory and applications. Many statistical models are naturally made up of distributions whose coordinates satisfy polynomial equations. Viewing these models as algebraic varieties, we may use tools from algebraic geometry to gain additional insight into their properties. Alternatively, we may revisit familiar algebraic and geometric objects in the setting of probability distributions.

In the first part of this thesis, I develop the theory of logarithmic Voronoi cells. These are convex sets used to divide experimental data based on which point in the model each sample most likely came from. For finite models, linear models, toric models, and models of maximum likelihood degree one, I prove that these sets are polytopes and characterize them combinatorially. I then use their structure to maximize information divergence to linear and toric models. For the latter family, I present a new algorithm for computing maximizers using vertices of logarithmic Voronoi polytopes. For non-polytopal logarithmic Voronoi cells, I develop a method to compute them via the framework of numerical algebraic geometry.

The second and third parts of this dissertation focus on conditional independence. In the second part, I study context-specific independence and introduce the family of decomposable CSmodels. I prove that these models mirror many of the algebraic and combinatorial properties that characterize decomposable graphical models, and hence they are good candidates for decomposable models in the context-specific setting. I give the strongest possible algebraic characterization of decomposable CSmodels by describing their prime ideals. In the third part, I focus on nonparametric algebraic statistics. I study dimensions, defining polynomials, and degrees of the moment varieties of conditionally independent mixture distributions. The last chapter features both symbolic and numerical computational methods.