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34
An Introduction to MCMC for Machine Learning
, 2003
"... This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of ..."
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Cited by 247 (2 self)
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This purpose of this introductory paper is threefold. First, it introduces the Monte Carlo method with emphasis on probabilistic machine learning. Second, it reviews the main building blocks of modern Markov chain Monte Carlo simulation, thereby providing and introduction to the remaining papers of this special issue. Lastly, it discusses new interesting research horizons.
Controlled MCMC for Optimal Sampling
, 2001
"... this paper we develop an original and general framework for automatically optimizing the statistical properties of Markov chain Monte Carlo (MCMC) samples, which are typically used to evaluate complex integrals. The MetropolisHastings algorithm is the basic building block of classical MCMC methods ..."
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Cited by 37 (6 self)
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this paper we develop an original and general framework for automatically optimizing the statistical properties of Markov chain Monte Carlo (MCMC) samples, which are typically used to evaluate complex integrals. The MetropolisHastings algorithm is the basic building block of classical MCMC methods and requires the choice of a proposal distribution, which usually belongs to a parametric family. The correlation properties together with the exploratory ability of the Markov chain heavily depend on the choice of the proposal distribution. By monitoring the simulated path, our approach allows us to learn "on the fly" the optimal parameters of the proposal distribution for several statistical criteria. Keywords: Monte Carlo, adaptive MCMC, calibration, stochastic approximation, gradient method, optimal scaling, random walk, Langevin, Gibbs, controlled Markov chain, learning algorithm, reversible jump MCMC. 1. Motivation 1.1. Introduction Markov chain Monte Carlo (MCMC) is a general strategy for generating samples x i (i = 0; 1; : : :) from complex highdimensional distributions, say defined on the space X ae R nx , from which integrals of the type I (f) = Z X f (x) (x) dx; can be calculated using the estimator b I N (f) = 1 N + 1 N X i=0 f (x i ) ; provided that the Markov chain produced is ergodic. The main building block of this class of algorithms is the MetropolisHastings (MH) algorithm. It requires the definition of a proposal distribution q whose role is to generate possible transitions for the Markov chain, say from x to y, which are then accepted or rejected according to the probabilityy ff (x; y) = min ae 1; (y) q (y; x) (x) q (x; y) oe : The simplicity and universality of this algorithm are both its strength and weakness. The choice of ...
Bayesian and likelihood methods for fitting multilevel models with complex level1 variation
, 2002
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Multilevel Modelling of Medical Data
, 2001
"... This tutorial presents an overview of multilevel or hierarchical data modelling and its applications in medicine. A description of the basic model for nested data is given and it is shown how this can be extended to fit flexible models for repeated measures data and more complex structures involving ..."
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Cited by 12 (0 self)
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This tutorial presents an overview of multilevel or hierarchical data modelling and its applications in medicine. A description of the basic model for nested data is given and it is shown how this can be extended to fit flexible models for repeated measures data and more complex structures involving cross classifications and multiple membership patterns within the software package MLwiN. A variety of response types is covered and both frequentist and Bayesian estimation methods are described.
CrossClassified and Multiple Membership Structures in Multilevel Models: An Introduction and Review: Research Report Number 791
, 2006
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Covariance Components Models for Longitudinal Family Data
 International Journal of Epidemiology
, 2005
"... A longitudinal family study is an epidemiological design that involves repeated measurements over time in a sample that includes families. Such studies, that may also include relative pairs and unrelated individuals, allow closer investigation of not only the factors that cause a disease to arise, b ..."
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Cited by 5 (0 self)
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A longitudinal family study is an epidemiological design that involves repeated measurements over time in a sample that includes families. Such studies, that may also include relative pairs and unrelated individuals, allow closer investigation of not only the factors that cause a disease to arise, but also the genetic and environmental determinants that modulate the subsequent progression of that disease. Knowledge of such determinants may pay high dividends in terms of prognostic assessment and in the development of new treatments that may be tailored to the prognostic profile of individual patients. Unfortunately longitudinal family studies are difficult to analyse. They conflate the complex withinfamily correlation structure of a crosssectional family study with the correlation over time that is intrinsic to longitudinal repeated measures. Here we describe an approach to analysis that is relatively straightforward to implement, yet is flexible in its application. It represents a natural extension of a Gibbssamplingbased approach to the analysis of crosssectional family studies that we have described previously. The approach can be applied to pedigrees of
Modeling heterogeneity in relationships between initial status and rates of change: Treating latent variable regression coefficients as random coefficients in a threelevel hierarchical model. Accepted for publication
 in Journal of Educational and Behavioral Statistics
, 2006
"... In studies of change in education and numerous other fields, interest often centers on how differences in the status of individuals at the start of a time period of substantive interest relate to differences in subsequent change. In this report we present a fully Bayesian approach to estimating thre ..."
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Cited by 4 (1 self)
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In studies of change in education and numerous other fields, interest often centers on how differences in the status of individuals at the start of a time period of substantive interest relate to differences in subsequent change. In this report we present a fully Bayesian approach to estimating threelevel hierarchical models in which latent variable regression coefficients capturing the relationship between initial status and rates of change within each of J schools (Bwj, j = 1, …, J) are treated as varying across schools. Through analyses of data from the Longitudinal Study of American Youth, we show how modeling differences in Bwj as a function of school characteristics can broaden the kinds of questions we can address in school effects research. We illustrate the possibility of conducting sensitivity analyses employing t distributional assumptions at each level of
Decomposition of prediction error in multilevel models
, 2002
"... We present a decomposition of prediction error for the multilevel model in the context of predicting a future observable y∗j in the jth group of a hierarchical dataset. The multilevel prediction rule is used for prediction and the components of prediction error are estimated via a simulation study t ..."
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Cited by 3 (1 self)
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We present a decomposition of prediction error for the multilevel model in the context of predicting a future observable y∗j in the jth group of a hierarchical dataset. The multilevel prediction rule is used for prediction and the components of prediction error are estimated via a simulation study that spans the various combinations of level1 (individual) and level2 (group) sample sizes and different intraclass correlation values. Additionally, analytical results present the increase in predicted mean square error (PMSE) with respect to prediction error bias. The components of prediction error provide information with respect to the cost of parameter estimation versus data imputation for predicting future values in a hierarchical data set. Specifically, the cost of parameter estimation is very small compared to data imputation.
Thermodynamic limit of the GinzburgLandau equations
 Nonlinearity
, 1994
"... In this paper we consider the problem of nding a generic algorithm for applying Markov chain Monte Carlo (MCMC) estimation procedures to statistical models that include variance matrices with additional parameter constraints. We consider separately the case of additional constraints across variance ..."
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Cited by 2 (1 self)
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In this paper we consider the problem of nding a generic algorithm for applying Markov chain Monte Carlo (MCMC) estimation procedures to statistical models that include variance matrices with additional parameter constraints. We consider separately the case of additional constraints across variance matrices and review existing work on the case of additional parameter constraints within a variance matrix. We propose two simple singlesite updating random walk Metropolis algorithms which have the advantage of generality in that they can be applied to virtually all scenarios. We then consider four applications where these methods can be used in practice. We nally consider when such singlesite algorithms break down and discuss brie y multiple site alternatives.
How and when does complex reasoning occur? Empirically driven development of a learning progression focused on complex reasoning about biodiversity
 Journal of Research in Science Teaching
, 2009
"... Abstract: In order to compete in a global economy, students are going to need resources and curricula focusing on critical thinking and reasoning in science. Despite awareness for the need for complex reasoning, American students perform poorly relative to peers on international standardized tests m ..."
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Abstract: In order to compete in a global economy, students are going to need resources and curricula focusing on critical thinking and reasoning in science. Despite awareness for the need for complex reasoning, American students perform poorly relative to peers on international standardized tests measuring complex thinking in science. Research focusing on learning progressions is one effort to provide more coherent science curricular sequences and assessments that can be focused on complex thinking about focal science topics. This article describes an empirically driven, fivestep process to develop a 3year learning progression focusing on complex thinking about biodiversity. Our efforts resulted in empirical results and work products including: (1) a revised definition of learning progressions, (2) empirically driven, 3year progressions for complex thinking about biodiversity, (3) an application of statistical approaches for the analysis of learning progression products, (4) Hierarchical Linear Modeling results demonstrating significant student achievement on complex thinking about biodiversity, and (5) Growth Model results demonstrating strengths and weaknesses of the first version of our curricular units. The empirical studies present information to inform both curriculum and assessment development. For curriculum development, the role of learning progressions as templates for the development of organized sequences of curricular units focused on complex science is discussed. For assessment development, learning progressionguided assessments provide a greater range and amount of information that can more reliably discriminate