Slide 1DataGibbs Sampling For Logistic Data?AlgorithmImplementationInitial ProblemsSampler Output/DiagosticsSampler Output/DiagnosticsSampler Output/DiagnosticsSampler Output/DiagnosticsSampler Output/DiagnosticsWinBUGS ModelWinBUGS ModelWinBUGS Output: Beta0 (1,0)WinBUGS Output: Beta0 (1,1)WinBUGS Output: Beta0 (1,-2)ComparisonWinBUGS WinsResourcesA (poor) Gibbs Sampling Approach to Logistic RegressionKyle BogdanGrant BrownDataSimulated based on known values of parameters (one covariate, ‘dose’).‘rats’ given different dosages of imaginary chemical, 4 dose groups with 25 rats in each group.Data generated three times under different parameters, three chains used for each data set.Gibbs Sampling For Logistic Data?Traditionally, binomial likelihood, prior on logit.Full Conditionals have no coherent form.Attractive, however, because it eliminates the need to reject iterationsAlgorithmGroenewald and Mokgatlhe, 2005Create Uniform Latent Variables Based on Y[i,j] = 0, 1Draws from joint posterior of Betas and U[i,j]pi[i] = p(uniform(01) <= logit-1(Beta*x[i]))Written in R, refined in PythonVery inefficientDraw new parameter for each Y[i,j] at each iterationImplementation•Three datasets•Three chains per set•1 Million iterations per chain•Last 500k iterations sent to CODA•9m total iterations, 4.5 m analyzedInitial ProblemsSampler Output/DiagosticsSampler Output/DiagnosticsSampler Output/DiagnosticsSampler Output/DiagnosticsSampler Output/DiagnosticsWinBUGS ModelY[i,j]’s given binomial (instead of Bernoulli) likelihoodBetas regressed on logit of proportionLocally uniform priors on beta1 and beta2WinBUGS Modelmodel{for (i in 1:N){ r[i] ~ dbin(p[i], n[i]); logit(p[i]) <- (beta1 + beta2*(x[i] - mean(x[]))); r.hat[i] <- (p[i] * n[i]); }beta1 ~ dflat();beta2 ~ dflat();beta1nocenter <- beta1 - beta2*mean(x[]);}WinBUGS Output: Beta0 (1,0)WinBUGS Output: Beta0 (1,1)WinBUGS Output: Beta0 (1,-2)ComparisonWinBUGS WinsUses proportions instead of Individual Y[i,j]’sConvergence is BetterWinBUGS appears more precise (more trials needed)Also, much faster.ResourcesGroenewald, Pieter C.N., and Lucky Mokgatlhe. "Bayesian computation for logistic regression.“ Computational Statistics & Data Analysis 48 (2005): 857-68. Science Direct. Elsevier. Web. <http://www.sciencedirect.com/>.Professor
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