Slide 1Introduction to MCMCSome VIPBG examplesCritical Sources for Code MRC BUGS Project OpenBUGSWhy bother?PayoffGenerally:Some applicationsThis introductionSlide 10Basic Ideas(Maximum) LikelihoodProblem with MLMaximum Likelihood (ML) “Thinks” (theoretically) about parameters and data separately: P(data|parameters) “Thinks” (practically) of integration, searching and finding confidence intervals as separate numerical problems (quadrature, e.g. Newton-Raphson, numerical differentiation).Markov Chain Monte Carlo (MCMC, MC2) “Thinks” (theoretically) that there is no difference between parameters and data – seeks distribution of parameters given data – P(parameters|data) {Bayesian estimation} “Thinks” (practically) that integration, search and interval estimation constitute a single process addressed by a single unifying algorithm {Gibbs Sampling}“Parameter”Basic approach“Bayesian”“Monte Carlo”“Markov Chain”“Gibbs Sampler”WinBUGSProblemsBottom lineTodayIntroduction to MCMC“Bayesian Inference Using Gibbs Sampling”Lindon EavesVIPBG, Spring 2011.Some VIPBG examplesEaves L and Erkanli A: Markov Chain Monte Carlo approaches to analysis of genetic and environmental components of human developmental change and GxE interaction, Behav Genetics May;33(3):279-98, 2003.Eaves L, Erkanli A, Silberg J, Angold A, Maes HH, Foley D (2005). Application of Bayesian Inference using Gibbs Sampling to Item-Response Theory Modelingof Multi-Symptom Genetic Data. Behav. Genet. Nov;35(6):765-80.Eaves L, Silberg J, and Erkanli A.: Resolving multiple epigenetic pathways to adolescent depression. J. Child Psychol Psychiatry Oct;44(7):1006-14, 2003.Eaves L, Silberg J, Foley D, Bulik C, Maes H. Erkanli A, Angold A, Costello EJ, Worthman C: Genetic and environmental influences on the relative timing of pubertal change. Twin Res. Oct;7(5);471-81, 2004.Franz CE, York TP, Eaves LJ, Mendoza SP, Hauger RL, Hellhammer DH, Jacobson KC, Levine S, Lupien SJ, Lyons MJ, Prom-Wormley E, Xian H, Kremen WS. (2010) Genetic and Environmental Influences on Cortisol Regulation Across Days and Contexts in Middle-Aged Men. Behav Genet. In pressYork TP, Strauss JF 3rd, Neale MC, Eaves LJ. (2009) Estimating fetal and maternal genetic contributions to premature birth from multiparous pregnancy histories of twins using MCMC and maximum-likelihood approaches. Twin Res Hum Genet.12:333-42. Eaves LJ, Silberg JL. Developmental-genetic effects on level and change in childhood fears of twins during adolescence. J Child Psychol Psychiatry. 2008 Nov;49(11):1201-10. Wray NR, Coventry WL, James MR, Montgomery GW, Eaves LJ, Martin NG.Use of monozygotic twins to investigate the relationship between 5HTTLPR genotype, depression and stressful life events: an application of Item Response Theory. Novartis Found Symp. 2008;293:48-59; discussion 59-70.Critical Sources for CodeMRC BUGS ProjectOpenBUGShttp://www.mrc-bsu.cam.ac.uk/bugs/http://www.openbugs.info/w/DownloadsWhy bother?•Intellectual challenge. Different (“Bayesian”) way of thinking about statistics and statistical modeling.•“Model Liberation” – can do a bunch of cool stuff much more easily•Learn more about data for less computational effort•“Fast”Payoff•Estimate parameters of complex models•Obtain subject parameters (e.g. “genetic and environmental factor scores”) at no extra-cost•Obtain confidence intervals and other summary statistics (s.e’s, quantiles etc) at no extra cost.•Automatic imputation of missing data (“data augmentation”)•Fast (35 item, IRT in 500 twins with covariates takes about 1-2 hours on laptop).•Insight, flexibilityGenerally:Seems to help with models that require multi-dimensional integration to compute likelihood.Some applications•Non-linear latent variables (GxE interaction).•Multidimensional, multi-category, multi-item IRT in twins.•Genetic effects on developmental change in multiple indicators of puberty (latent growth curves).•Hierarchical mixed models for fixed and random effects of G, E and GxE in multi-symptom (“IRT”) twin data – complex data structures (e.g. batch effects)•Genetic survival models•Mixture modelsThis introduction•Introduce ideas•Practice use of WinBUGS•Run some basic examples•Look at application to genetic IRT•Other stuff?Some references:Gilks WR, Richardson S, Spiegelhalter DJ (1996) Markov Chain Monte Carlo in Practice. Boca Raton,Chapman & Hall, Gelman A, Carlin JB, Stern HS, Rubin DB. (2004)Bayesian Data Analysis (2nd Ed,) Boca Raton,Chapman & Hall.Spiegelhalter DJ, Thomas A, Best N, Lunn D. (2004). WinBUGS UserManual Version 1.4.1. Cambridge, England. MRC BUGS project. [Downloaded with WinBUGS – also Examples Vols. I and II]Maris, G and Bechger, T.M. (2005). An Introduction to the DA-T Gibbs Sampler for the Two-Parameter Logistic (2PL) Model andBeyond. Psicol´ogica: 26, 327-352.http://www.uv.es/~revispsi/articulos2.05/8-MARIS.pdfBasic Ideas•Bayesian Estimation (vs. ML)•“Monte Carlo”•“Markov Chain”•Gibbs sampler(Maximum) Likelihood•Compute (log-) likelihood of getting data given values of model parameters and assumed distribution•Search for parameter values that maximize likelihood (“ML” estimates)•Compare models by comparing likelihoods•Obtain confidence intervals by contour plots (i.e. repeated ML conditional on selected parameters)•Obtain s.e.’s by differentiating LProblem with ML•Many models require integration over values of latent variables (e.g. non-linear random effects)•Integrate to evaluate each likelihood and derivatives for each parameter•“Expensive” when number of dimensions is large (?days), especially for confidence intervals.Maximum Likelihood (ML)“Thinks” (theoretically) about parameters and data separately: P(data|parameters) “Thinks” (practically) of integration, searching and finding confidence intervals as separate numerical problems (quadrature, e.g. Newton-Raphson, numerical differentiation).Markov Chain Monte Carlo (MCMC, MC2)“Thinks” (theoretically) that there is no difference between parameters and data – seeks distribution of parameters given data – P(parameters|data) {Bayesian estimation}“Thinks” (practically) that integration, search and interval estimation constitute a single process addressed by a single unifying algorithm {Gibbs Sampling}“Parameter”Anything that isn’t data: means, components of variance,
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