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12
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...
Estimating Normalizing Constants and Reweighting Mixtures in Markov Chain Monte Carlo
, 1994
"... Markov chain Monte Carlo (the MetropolisHastings algorithm and the Gibbs sampler) is a general multivariate simulation method that permits sampling from any stochastic process whose density is known up to a constant of proportionality. It has recently received much attention as a method of carrying ..."
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Cited by 40 (0 self)
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Markov chain Monte Carlo (the MetropolisHastings algorithm and the Gibbs sampler) is a general multivariate simulation method that permits sampling from any stochastic process whose density is known up to a constant of proportionality. It has recently received much attention as a method of carrying out Bayesian, likelihood, and frequentist inference in analytically intractable problems. Although many applications of Markov chain Monte Carlo do not need estimation of normalizing constants, three do: calculation of Bayes factors, calculation of likelihoods in the presence of missing data, and importance sampling from mixtures. Here reverse logistic regression is proposed as a solution to the problem of estimating normalizing constants, and convergence and asymptotic normality of the estimates are proved under very weak regularity conditions. Markov chain Monte Carlo is most useful when combined with importance reweighting so that a Monte Carlo sample from one distribution can be used fo...
Hypothesis Testing and Model Selection Via Posterior Simulation
 In Practical Markov Chain
, 1995
"... Introduction To motivate the methods described in this chapter, consider the following inference problem in astronomy (Soubiran, 1993). Until fairly recently, it has been believed that the Galaxy consists of two stellar populations, the disk and the halo. More recently, it has been hypothesized tha ..."
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Cited by 24 (1 self)
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Introduction To motivate the methods described in this chapter, consider the following inference problem in astronomy (Soubiran, 1993). Until fairly recently, it has been believed that the Galaxy consists of two stellar populations, the disk and the halo. More recently, it has been hypothesized that there are in fact three stellar populations, the old (or thin) disk, the thick disk, and the halo, distinguished by their spatial distributions, their velocities, and their metallicities. These hypotheses have different implications for theories of the formation of the Galaxy. Some of the evidence for deciding whether there are two or three populations is shown in Figure 1, which shows radial and rotational velocities for n = 2; 370 stars. A natural model for this situation is a mixture model with J components, namely y i = J X j=1 ae j
MONTE CARLO LIKELIHOOD INFERENCE FOR MISSING DATA MODELS
"... We describe a Monte Carlo method to approximate the maximum likelihood estimate (MLE), when there are missing data and the observed data likelihood is not available in closed form. This method uses simulated missing data that are independent and identically distributed and independent of the observe ..."
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Cited by 4 (2 self)
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We describe a Monte Carlo method to approximate the maximum likelihood estimate (MLE), when there are missing data and the observed data likelihood is not available in closed form. This method uses simulated missing data that are independent and identically distributed and independent of the observed data. Our Monte Carlo approximation to the MLE is a consistent and asymptotically normal estimate of the minimizer θ ∗ of the KullbackLeibler information, as both Monte Carlo and observed data sample sizes go to infinity simultaneously. Plugin estimates of the asymptotic variance are provided for constructing confidence regions for θ ∗. We give LogitNormal generalized linear mixed model examples, calculated using an R package. AMS 2000 subject classifications. Primary 62F12; secondary 65C05. Key words and phrases. Asymptotic theory, Monte Carlo, maximum likelihood, generalized
Discussion of the paper "Markov chains for exploring posterior distributions" by Luke Tierney
 Leipzig und Berlin
, 1994
"... this paper, which even before its appearance has done a valuable service in clarifying both theory and practice in this important area. For example, the discussion of combining strategies in Section 2.4 helped researchers break away from pure Gibbs sampling in 1991; it was, for example, part of the ..."
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Cited by 3 (0 self)
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this paper, which even before its appearance has done a valuable service in clarifying both theory and practice in this important area. For example, the discussion of combining strategies in Section 2.4 helped researchers break away from pure Gibbs sampling in 1991; it was, for example, part of the reasoning that lead to the "Metropoliscoupled" scheme of Geyer (1991) mentioned at the end of Section 2.3.3.
Method of Moments Using Monte Carlo Simulation
 Journal of Computational and Graphical Statistics
, 1995
"... We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the loglikelihood. We present a simple implementation using ..."
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Cited by 2 (1 self)
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We present a computational approach to the method of moments using Monte Carlo simulation. Simple algebraic identities are used so that all computations can be performed directly using simulation draws and computation of the derivative of the loglikelihood. We present a simple implementation using the NewtonRaphson algorithm, with the understanding that other optimization methods may be used in more complicated problems. The method can be applied to families of distributions with unknown normalizing constants and can be extended to leastsquares fitting in the case that the number of moments observed exceeds the number of parameters in the model. The method can be further generalized to allow "moments" that are any function of data and parameters, including as a special case maximum likelihood for models with unknown normalizing constants or missing data. In addition to being used for estimation, our method may be useful for setting the parameters of a Bayes prior distribution by spe...
Multipoint linkage analyses for disease mapping in extended pedigrees: A Markov chain Monte Carlo approach
, 2002
"... Multipoint linkage analyses ofgenetic data on extended pedigrees can involve exact computations which are infeasible. Markov chain Monte Carlo methods represent an attractive alternative, greatly extending the range of models and data sets for which analysis is practical. In this paper, several adva ..."
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Cited by 1 (1 self)
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Multipoint linkage analyses ofgenetic data on extended pedigrees can involve exact computations which are infeasible. Markov chain Monte Carlo methods represent an attractive alternative, greatly extending the range of models and data sets for which analysis is practical. In this paper, several advances in Markov chain Monte Carlo theory, namely joint updates of latent variables across loci and meioses, integrated proposals, MetropolisHastings restarts via sequential imputation and Rao Blackwellized estimators, are incorporated into a sampling strategy which mixes well and produces accurate results in real time. The methodology is demonstrated through its application to several data sets originating from a study of earlyonset Alzheimer's disease in families of VolgaGerman ethnic origin.
Bayes Factors via Serial Tempering
, 2010
"... Let M be a finite or countable set of models (here we only deal with finite M but Bayes factors make sense for countable M). For each model m ∈ M we have the prior probability of the model pri(m). It does not ..."
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Cited by 1 (0 self)
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Let M be a finite or countable set of models (here we only deal with finite M but Bayes factors make sense for countable M). For each model m ∈ M we have the prior probability of the model pri(m). It does not
Maximum Likelihood Estimation for a Parametric Dispersal Model for Haploid Organisms
, 1999
"... Understanding the fine scale genetic structure of a population is important in conservation genetics, ecology, and gauging the impact of the release of genetically engineered or nonnative organisms. One of the important factors contributing to this genetic structure is dispersal: how far and in wha ..."
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Understanding the fine scale genetic structure of a population is important in conservation genetics, ecology, and gauging the impact of the release of genetically engineered or nonnative organisms. One of the important factors contributing to this genetic structure is dispersal: how far and in what direction does an individual travel between its birth and the time it reproduces? I present a method for inferring the maximum likelihood estimate of a parametric distribution for this distance in a discrete generation haploid population, using genetic data and binned spatial information at two consecutive generations. Likelihoods are obtained by using Monte Carlo methods to approximate a sum over a large number of discrete latent variables. I examine the effects on inference of bin size, initial population configuration, and offspring distribution. Initial population configuration in particular plays a large role in how much information a data set provides. Supported by a National Science Foundation Graduate Research Fellowship