# Algebras of gaussian linear information

23.06.2009

XVI + 354 p

Thèse de doctorat: Université de Fribourg, 2009

English German Gaussian linear information arises in many real-world models of the natural and social sciences. The Gaussian distribution has turned out to appropriately represent uncertainty in many linear models. The main goal of this thesis is to describe and to compare different algebras of Gaussian linear information: Corresponding elements and operations in the various algebras are revealed and the respective computational advantages are highlighted. In order to make large models computationally tractable, they have to be decomposed into independent factors by exploiting sparsity. For such factorisations, valuation algebras provide a general, abstract framework for local computations. A valuation algebra is a two-sorted algebra with three operations: valuations (which may be seen as pieces of information) refer to a domain of interest; valuations can be marginalised (focussed) to a domain of interest, and they may be combined (aggregated). Generic message-passing schemes can be used to answer projection problems. Many problems in applications can be reduced to a projection problem: diagnostic estimation, prediction, filtering, smoothing. For instance, Gaussian densities form a valuation algebra: marginalisation is integration, and combination is multiplication (plus renormalisation). Gaussian densities may be represented by Gaussian potentials or moment matrices, using either the concentration or the variance-covariance matrix, respectively. Here, marginalisation and combination are matrix operations. A conditional Gaussian density is the family of Gaussian densities obtained on the head variables by fixing a value for the tail variables. A conditional Gaussian density corresponds to a Gaussian density on the head variables plus a linear regression on the tail variables. Conditional Gaussian densities can be analysed in three ways: geometric, algebraic and analytic. General Gaussian linear systems lead to Gaussian hints by assumption-based inference. Gaussian hints have focal sets which are parallel linear manifolds of the same dimension in the parameter space. Combination corresponds to intersection of focal sets and marginalisation to projection of focal sets. Gaussian potentials correspond to Gaussian hints whose focal sets are all singletons. Gaussian potentials can be extended to a valuation algebra of quotients which are represented by pairs of Gaussian potentials. Conditional Gaussian densities can be represented in the so-called separative extension of Gaussian potentials. Since a conditional Gaussian density is a quotient function of two Gaussian densities, the concentration matrix in the exponent of the denominator can be subtracted from the concentration matrix in the exponent of the numerator. This leads to a new representation of symmetric Gaussian potentials whose pseudo-concentration matrix is only symmetric but not necessarily positive definite. The main result of these considerations is that different conditional Gaussian densities turn out to be linked to the same Gaussian hints (up to equivalence) if and only if the conditional Gaussian densities are equal up to a constant factor. In other words, Gaussian likelihood functions bear the full information contained in Gaussian hints. This explains why assumption-based reasoning on (over-)determined Gaussian linear systems reproduces the estimation results based on the maximum-likelihood principle. Variables may be linear combinations of other variables. This imposes linear restrictions on the parameter space. In the spirit of assumption-based reasoning, algorithms for inference, the combination and marginalisation are derived for symmetric Gaussian potentials with deterministic equations. Finally, it is shown how Gaussian linear systems can be expressed in the language Abel. Queries on a complex Gaussian linear system can be answered in the Abel system. Several examples illustrate the new approach of symmetric Gaussian potentials.
Faculty
Faculté des sciences et de médecine
Department
Département d'Informatique
Language
• English
Classification
Computer science and technology
Notes
• Ressource en ligne consultée le 13.01.2010