This thesis deals with the problem of estimating statistical distributions from data. In the first part, the method of moments is used in combination with computational algebraic techniques in order to estimate parameters coming from local Dirac mixtures and their convolutions. The second part focuses on the nonparametric setting, in particular on combinatorial and algebraic aspects of the estimation of log-concave distributions.

In this thesis, we address problems from two topics of applied mathematics: linear integer programming and polyhedral computation. Linear integer programming concerns solving optimisation problems to maximise a linear cost function over the set of integer points in a polyhedron. Polyhedral computation is concerned with algorithms for computing different properties of convex polyhedra. First, we explore the theory and computation of Gröbner bases and Markov bases for linear integer programming. Second, we investigate and improve an algorithm from polyhedral computation that converts between different representations of cones and polyhedra. A Markov basis is a set of integer vectors such that we can move between any two feasible solutions of an integer program by adding or subtracting vectors in the Markov basis while never moving outside the set of feasible solutions. Markov bases are mainly used in algebraic statistics for sampling from a set of feasible solutions. The major contribution of this thesis is a fast algorithm for computing Markov bases, which we used to solve a previously intractable computational challenge. Gröbner basis methods are exact local search approaches for solving integer programs. We present a Gröbner basis approach that can use the structure of an integer program in order to solve it more efficiently. Gröbner basis methods are interesting mainly from a purely theoretical viewpoint, but they are also interesting because they may provide insight into why some classes of integer programs are difficult to solve using standard techniques and because someday they may be able to solve these difficult problems. Computing the properties of convex polyhedra is useful for solving problems within different areas of mathematics such as linear programming, integer programming, combinatorial optimisation, and computational geometry. We investigate and improve an algorithm for converting between a generator representation of a cone or polyhedron and a constraint representation of the cone or polyhedron and vice versa. This algorithm can be extended to compute circuits of matrices, which are used in computational biology for metabolic pathway analysis.

This thesis is about statistical models and algebraic varieties. Algebraic Statistics unites these two concepts, turning algebraic structure into statistical insight. Featured here are three types of models that have such an algebraic structure. Linear Gaussian covariance models are continuous models which are simple to define but hard to analyze. We compute their maximum likelihood degree in dimension two and find it equal to $2n-3$ generically if the model has $n$ covariates. Discrete models with rational MLE are those discrete models for which likelihood estimation is easiest. We characterize them geometrically by building on the work of Huh and Kapranov on Horn uniformization. Algebraic manifolds are a more general kind of object which is used to encode continuous data. We introduce a new method for computing integrals and sampling from distributions on them, based on intersecting with random linear spaces. A brief report on mathematics in the sciences featuring case studies from soil ecology and nonparametric statistics closes the thesis.

The Maximum Likelihood Degree (ML degree) of a statistical model is the number of complex critical points of the likelihood function. In this thesis we study this on Colored Gaussian Graphical Models, classifying the ML degree of colored graphs of order up to three. We do this by calculating the rational function degree of the gradient of the log- likelihood. Moreover we find that coloring a graph can lower the ML degree. Finally we calculate solutions to the homaloidal partial differential equation developed by Améndola et al. The code developed for these calculations can be used on graphs of higher orders. / Maximum likelihood-graden (ML-graden) för en statistisk modell är antalet komplexa kritiska punkter för likelihoodfunktionen. I denna avhandling studerar vi detta på färgade Gaussiska grafiska modeller och klassificerar ML-graden för färgade grafer av ordning upp till tre. Detta görs genom att beräkna den rationella funktionsgraden för gradienten av logaritmen av likelihoodfunktionen. Dessutom finner vi att ML-graden av en graf kan minskas genom att färgläggas. Slutligen beräknar vi lösningar till den homaloidala partiella differentialekvationen utvecklad av Améndola et al. Den kod som utvecklats för dessa beräkningar kan användas på grafer av högre ordning.

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.

In the thesis "On Boundaries of Statistical Models" problems related to a description of probability distributions with zeros, lying in the boundary of a statistical model, are treated. The distributions considered are joint distributions of finite collections of finite discrete random variables. Owing to this restriction, statistical models are subsets of finite dimensional real vector spaces. The support set problem for exponential families, the main class of models considered in the thesis, is to characterize the possible supports of distributions in the boundaries of these statistical models. It is shown that this problem is equivalent to a characterization of the face lattice of a convex polytope, called the convex support. The main tool for treating questions related to the boundary are implicit representations. Exponential families are shown to be sets of solutions of binomial equations, connected to an underlying combinatorial structure, called oriented matroid. Under an additional assumption these equations are polynomial and one is placed in the setting of commutative algebra and algebraic geometry. In this case one recovers results from algebraic statistics. The combinatorial theory of exponential families using oriented matroids makes the established connection between an exponential family and its convex support completely natural: Both are derived from the same oriented matroid. The second part of the thesis deals with hierarchical models, which are a special class of exponential families constructed from simplicial complexes. The main technical tool for their treatment in this thesis are so called elementary circuits. After their introduction, they are used to derive properties of the implicit representations of hierarchical models. Each elementary circuit gives an equation holding on the hierarchical model, and these equations are shown to be the "simplest", in the sense that the smallest degree among the equations corresponding to elementary circuits gives a lower bound on the degree of all equations characterizing the model. Translating this result back to polyhedral geometry yields a neighborliness property of marginal polytopes, the convex supports of hierarchical models. Elementary circuits of small support are related to independence statements holding between the random variables whose joint distributions the hierarchical model describes. Models for which the complete set of circuits consists of elementary circuits are shown to be described by totally unimodular matrices. The thesis also contains an analysis of the case of binary random variables. In this special situation, marginal polytopes can be represented as the convex hulls of linear codes. Among the results here is a classification of full-dimensional linear code polytopes in terms of their subgroups. If represented by polynomial equations, exponential families are the varieties of binomial prime ideals. The third part of the thesis describes tools to treat models defined by not necessarily prime binomial ideals. It follows from Eisenbud and Sturmfels'' results on binomial ideals that these models are unions of exponential families, and apart from solving the support set problem for each of these, one is faced with finding the decomposition. The thesis discusses algorithms for specialized treatment of binomial ideals, exploiting their combinatorial nature. The provided software package Binomials.m2 is shown to be able to compute very large primary decompositions, yielding a counterexample to a recent conjecture in algebraic statistics.