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17
On the Convergence of Monte Carlo Maximum Likelihood Calculations
 Journal of the Royal Statistical Society B
, 1992
"... Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the ..."
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Cited by 59 (3 self)
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Monte Carlo maximum likelihood for normalized families of distributions (Geyer and Thompson, 1992) can be used for an extremely broad class of models. Given any family f h ` : ` 2 \Theta g of nonnegative integrable functions, maximum likelihood estimates in the family obtained by normalizing the the functions to integrate to one can be approximated by Monte Carlo, the only regularity conditions being a compactification of the parameter space such that the the evaluation maps ` 7! h ` (x) remain continuous. Then with probability one the Monte Carlo approximant to the log likelihood hypoconverges to the exact log likelihood, its maximizer converges to the exact maximum likelihood estimate, approximations to profile likelihoods hypoconverge to the exact profile, and level sets of the approximate likelihood (support regions) converge to the exact sets (in Painlev'eKuratowski set convergence). The same results hold when there are missing data (Thompson and Guo, 1991, Gelfand and Carlin, 19...
Blocking Gibbs Sampling for Linkage Analysis in Large Pedigrees with Many Loops
 AMERICAN JOURNAL OF HUMAN GENETICS
, 1996
"... 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 complexi ..."
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Cited by 24 (2 self)
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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.
Estimation of conditional multilocus gene identity among relatives
 STATISTICS IN MOLECULAR BIOLOGY AND GENETICS: SELECTED PROCEEDINGS OF A 1997 JOINT AMSIMSSIAM SUMMER CONFERENCE ON STATISTICS IN MOLECULAR BIOLOGY', VOL. 33 OF IMS LECTURE NOTEMONOGRAPH SERIES, INSTITUTE OF MATHEMATICAL STATISTICS
, 1999
"... Genetic Analysis Workshop 10 identified five key factors contributing to the resolution of the genetic factors affecting complex traits. These include analysis with multipoint methods, use of extended pedigrees, and selective sampling of pedigrees. By sampling the affected individuals in an extended ..."
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Cited by 12 (2 self)
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Genetic Analysis Workshop 10 identified five key factors contributing to the resolution of the genetic factors affecting complex traits. These include analysis with multipoint methods, use of extended pedigrees, and selective sampling of pedigrees. By sampling the affected individuals in an extended pedigree, we obtain individuals who have an increased probability of sharing genes identical by descent (IBD) at marker loci that are linked to the trait locus or loci. Given marker data on specified members of a pedigree, the conditional IBD status among relatives can be assessed, but exact computation is often impractical for multiple linked markers on complex pedigrees. The use of Markov chain Monte Carlo (MCMC) methods greatly extends the range of models and data sets for which analysis is computationally feasible. Many forms of MCMC have now been implemented in the context of genetic analysis. Here we propose a new sampler, which takes as latent variables the segregation indicators at marker loci, and jointly updates all indicators corresponding to a given meiosis. The sampler has good mixing properties. Questions of irreducibility are also addressed.
Multilocus linkage analysis by blocked Gibbs sampling
 Statistics and Computing
, 2000
"... The problem of multilocus linkage analysis is expressed as a graphical model, making explicit a previously implicit connection, and recent developments in the field are described in this context. A novel application of blocked Gibbs sampling for Bayesian networks is developed to generate inheritance ..."
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Cited by 9 (0 self)
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The problem of multilocus linkage analysis is expressed as a graphical model, making explicit a previously implicit connection, and recent developments in the field are described in this context. A novel application of blocked Gibbs sampling for Bayesian networks is developed to generate inheritance matrices from an irreducible Markov chain. This is used as the basis for reconstruction of historical meiotic states and approximate calculation of the likelihood function for the location of an unmapped genetic trait. We believe this to be the only approach that currently makes fully informative multilocus linkage analysis possible on large extended pedigrees.
Statistical Image Analysis and Confocal Microscopy
, 2001
"... Images are the source of information in many areas of scientific enquiry. A common objective in these applications is reconstruction of the true scene from a degraded image. When objects in the image can be described parametrically, reconstruction can proceed by fitting a high level image model. In ..."
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Cited by 4 (1 self)
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Images are the source of information in many areas of scientific enquiry. A common objective in these applications is reconstruction of the true scene from a degraded image. When objects in the image can be described parametrically, reconstruction can proceed by fitting a high level image model. In this article we consider the analysis of confocal fluorescence microscope images of cells in an area of cartilage growth. Biological questions posed by the experimenters concern the nature of the cells in the image and changes in their properties with time. Our model of the imaging process is based on a detailed analysis of the data. We treat the true scene as a realisation of a marked point process, incorporating this as the highlevel prior model in a Bayesian analysis. Inference is by simulation using reversible jump versions of Markov chain Monte Carlo (MCMC) algorithms which can handle the varying dimension of the image description arising from an unknown number of cells, each with its own parameters.
Problems with the Determination of the Noncommunicating Classes for MCMC Applications in Pedigree Analysis
, 1998
"... Exact calculations for probabilities on complex pedigrees are computationally intensive and very often infeasible. Markov chain Monte Carlo methods are frequently used to approximate probabilities and likelihoods of interest. However, when a locus with more than two alleles is considered, the ..."
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Cited by 4 (2 self)
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Exact calculations for probabilities on complex pedigrees are computationally intensive and very often infeasible. Markov chain Monte Carlo methods are frequently used to approximate probabilities and likelihoods of interest. However, when a locus with more than two alleles is considered, the underlying Markov chain is not guaranteed to be irreducible and the results of such analyses are unreliable. A method for finding the noncommunicating classes of the Markov chain would be very useful in designing algorithms that can jump between these classes. In this paper we will examine some existing work on this problem and point out its limitations. We will also 1 comment on the difficulty of developing a useful algorithm. Keywords: Complex pedigrees, reducibility, islands, Gibbs sampling 2 1 Introduction The computation of probabilities on pedigrees is an essential component in any analysis of genetic data on groups of related individuals. Such computations are relevant ...
Markov chain Monte Carlo methods for family trees using a parallel processor
 Statistics and Computing
, 1995
"... A 1024 cpu parallel computer is used to obtain simulated genotypes in the Tristan da Cunha pedigree using random local updating methods. A fourcolour theorem is invoked to justify simultaneous updating. Multiple copies of the program are run simultaneously. The results are used to infer the sou ..."
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Cited by 3 (0 self)
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A 1024 cpu parallel computer is used to obtain simulated genotypes in the Tristan da Cunha pedigree using random local updating methods. A fourcolour theorem is invoked to justify simultaneous updating. Multiple copies of the program are run simultaneously. The results are used to infer the source of the B allele of the ABO blood group that is present in the population. Keywords: Ancestral inference, Gibbs sampling, massively parallel, pedigree analysis. rjb@maths.bath.ac.uk y Corresponding author, alun@myriad.com 1 1 Introduction Calculating probabilities on family trees provides an interesting and useful application for Markov chain Monte Carlo methods. While exact methods of calculation are routinely used for pedigrees and genetic models of moderate complexity, these become infeasible when either the model, the pedigree or both are more complex. The computational time required grows exponentially with complexity. The exact methods used are variants of the peeling metho...
A Simple Method for Finding a Legal Configuration in Complex Bayesian Networks
, 1996
"... This paper deals with an important problem with large and complex Bayesian networks. Exact inference in these networks is simply infeasible due to the huge storage requirements of exact methods. Markov chain Monte Carlo methods, however, are able to deal with these large networks but to do this they ..."
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Cited by 1 (0 self)
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This paper deals with an important problem with large and complex Bayesian networks. Exact inference in these networks is simply infeasible due to the huge storage requirements of exact methods. Markov chain Monte Carlo methods, however, are able to deal with these large networks but to do this they require an initial legal configuration to set off the sampler. So far nondeterministic methods like forward sampling have often been used for this even though the forward sampler may take an eternity to come up with a legal configuration. In this paper a novel algorithm will be presented that allows finding a legal configuration in a general Bayesian network in polynomial time in almost all cases. The algorithm will not be proven deterministic but empirical results will document that this holds in most cases. Also, the algorithm will be justified by its simplicity and ease of implementation. Keywords: Bayesian network, junction tree, pedigree analysis, Markov chain Monte Carlo, Gibbs samp...
Bayesian Computation and Stochastic Systems
, 1995
"... advantages of a Bayesian approach. Whilst of course we recognize the importance and intellectual standing of the long debate about philosophies of inference at a more fundamental level, nevertheless it is surely true that some of the main historical objections to Bayesian inference have included th ..."
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advantages of a Bayesian approach. Whilst of course we recognize the importance and intellectual standing of the long debate about philosophies of inference at a more fundamental level, nevertheless it is surely true that some of the main historical objections to Bayesian inference have included the difficulty of computation, the need to approximate, the necessity to use stylized priors, and the inability to assess the impact of arbitrary assumptions in prior specifications. MCMC answers these objections amazingly well, and indeed also allows one to perturb the likelihood function. For those of us who were closet Bayesians, or at least are openminded enough to discover what the paradigm can provide, MCMC does remove reasons not to be Bayesian. Geyer's claim that similar progress has been made in likelihood inference is surely grossly overstated. Integration is central to the Bayesian paradigm but runs into problems for almost any moderately complicated
J. Dairy Sci. 85:1623–1629 © American Dairy Science Association, 2002. Technical Note: Determining Peeling Order Using Sparse Matrix Algorithms 1
"... To study the effect of individual genes by segregation or linkage analyses, the likelihood of the model needs to be evaluated. The likelihood can be computed efficiently using the ElstonStewart algorithm. This algorithm involves summing over the unobserved genotypes in the pedigree, which is called ..."
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To study the effect of individual genes by segregation or linkage analyses, the likelihood of the model needs to be evaluated. The likelihood can be computed efficiently using the ElstonStewart algorithm. This algorithm involves summing over the unobserved genotypes in the pedigree, which is called peeling. An important aspect of this algorithm is to determine the order of peeling to maximize efficiency. This paper shows how determining peeling order is related to a problem in solving systems of symmetric sparse linear equations. It also shows how algorithms developed to efficiently solve those systems, can be used to determine the optimal order of peeling in the ElstonStewart algorithm. (Key words: peeling order, sparse matrices)