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.
| Identifer | oai:union.ndltd.org:DRESDEN/oai:qucosa.de:bsz:15-qucosa-37952 |
| Date | 24 June 2010 |
| Creators | Kahle, Thomas |
| Contributors | Universität Leipzig, Fakultät für Mathematik und Informatik, Max-Planck-Institut für Mathematik in den Naturwissenschaften,, Dr. Nihat Ay, Prof. Dr. Jürgen Jost, Prof. Dr. Bernd Sturmfels |
| Publisher | Universitätsbibliothek Leipzig |
| Source Sets | Hochschulschriftenserver (HSSS) der SLUB Dresden |
| Language | English |
| Detected Language | English |
| Type | doc-type:doctoralThesis |
| Format | application/pdf |
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