We introduce a methodology for performing approximate computations in very complex probabilistic systems (e.g. huge pedigrees). Our approach, called blocking Gibbs, combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The methodology is illustrated on a real-world problem involving a heavily inbred pedigree containing 20;000 individuals. We present results showing that blocking-Gibbs sampling converges much faster than plain Gibbs sampling for very complex problems.

We will apply the method of blocking Gibbs sampling to a problem of great importance and complexity -- linkage analysis. Blocking Gibbs combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The method is able to handle problems with very high complexity such as linkage analysis in large pedigrees with many loops; a task that no other known method is able to handle. New developments of the method are outlined, and it is applied to a highly complex linkage problem. Keywords: Bayesian network, junction tree, pedigree analysis, Markov chain Monte Carlo, Gibbs sampling, loops, inbreeding 1 Introduction For linkage analysis - the problem of estimating the relative positions of the genes on the chromosomes - many methods have been developed over recent years. Fast and exact methods for computation in Bayesian networks (e.g., pedigrees) (Cannings, Thompson & Skolnick 1976; Pearl 1986; Lauritzen & Spiegelhalter 1988; Shenoy & Shafer 1990; Lauri...

The efficiency of inference in both the Hugin and, most notably, the Shafer-Shenoy architectures can be improved by exploiting the independence relations induced by the incoming messages of a clique. That is, the message to be sent from a clique can be computed via a factorization of the clique potential in the form of a junction tree. In this paper we show that by exploiting such nested junction trees in the computation of messages both space and time costs of the conventional propagation methods may be reduced. The paper presents a structured way of exploiting the nested junction trees technique to achieve such reductions. The usefulness of the method is emphasized through a thorough empirical evaluation involving ten large real-world Bayesian networks and the Hugin inference algorithm. 1 INTRODUCTION Inference in Bayesian networks can be formulated as message passing in a junction tree corresponding to the network (Jensen, Lauritzen & Olesen 1990, Shafer & Shenoy 1990). More precis...

This paper describes a non-iterative, recursive method to compute the likelihood for a pedigree without loops, and hence an efficient way to compute genotype probabilities for every member of the pedigree. The method can be used with multiple mates and large sibships. Scaling is used in calculations to avoid numerical problems in working with large pedigrees.

(Under the direction of Robert C. Elston.) A multifactorial model for the segregation analysis of quantitative traits in pedigrees is presented. The model includes both polygenic and monogenic effects., The model also allows for two types of environmental correlation, a within sibship correlation and a within nuclear family correlation. Methods for the approximation of the true likelihood e for this model are presented. These models are derived from the methods for estimating the parameters of a mixture of normal distributions. The estimation methods are those equating moments, maximum likelihood and two methods of least squares estimation. One least squares method involves minimizing the sum of squared differences between the exact likelihood function and the approximation function; the

1981 Approved by: /)r

We introduce a methodology for performing approximate computations in very complex probabilistic systems (e.g. huge pedigrees). Our approach, called blocking Gibbs, combines exact local computations with Gibbs sampling in a way that complements the strengths of both. The methodology is illustrated on a real-world problem involving a heavily inbred pedigree containing 20;000 individuals. We present results showing that blocking-Gibbs sampling converges much faster than plain Gibbs sampling for very complex problems.