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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 ..."
Abstract

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...
Monte Carlo Likelihood Calculation for Identity by Descent Data
, 1999
"... Two individuals are identical by descent at a genetic locus if they share the same gene copy at that locus due to inheritance from a recent common ancestor. Identity by descent can be thought of as a continuous process along the genome that is the outcome of a highly structured hidden process. The c ..."
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Two individuals are identical by descent at a genetic locus if they share the same gene copy at that locus due to inheritance from a recent common ancestor. Identity by descent can be thought of as a continuous process along the genome that is the outcome of a highly structured hidden process. The complexity of the structure rules out direct analytic methods for calculating likelihoods in most situations, so that a Monte Carlo approach is required. This thesis presents an approach that applies to many models for the underlying genetic process of crossingover at meiosis. The method is applied to simulated data in order to examine the amount of information contained in identity by descent data about the true model for the crossingover process and about the true relationship between the two individuals from whom the data derive. Much of the work is done with idealized continuous identity by descent data, but an extension to the Monte Carlo method is developed that allows analysis of real data. Real data consist of identity (not necessarily by descent) of gene copies at discrete locations along the genome. The method is applied to relationship inference analysis of a real data set.