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Enhancing Bayesian Inference with BUGS and MCMCglmm for Hierarchical Models

Explore the power of Bayesian inference using BUGS and the MCMCglmm package in R. This guide outlines how to build hierarchical models, set prior distributions, and assess convergence through MCMC diagnostics. Learn the essentials of specifying informative priors, sampling from posterior distributions, and interpreting results with credible intervals. The practical examples demonstrate the modeling process and diagnostics, fostering a deeper understanding of Bayesian methods for complex data structures.

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Enhancing Bayesian Inference with BUGS and MCMCglmm for Hierarchical Models

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Presentation Transcript

  1. I like Bayesian inference…how do I use it?

  2. BUGS links Model & Code model{ for(i in 1:N){ isigma[i] <- 1/sigma[i] d[i] ~ dnorm(theta[i], isigma[i]) theta[i] ~ dnorm(mu, itau) } #priors mu ~ dnorm(0, 1e-06) tau ~ dunif(0,1000) itau <- 1/tau }

  3. Hierarchical Model in BUGS model{ for(i in 1:N){ isigma[i] <- 1/sigma[i] d[i] ~ dnorm(phi[i], isigma[i]) phi[i] ~ dnorm(theta[Study[i]], iomega[Study[i]]) } for(j in 1:K){ theta[j] ~ dnorm(mu, itau) } #priors mu ~ dnorm(0, 1e-06) tau ~ dunif(0,1000) itau <- 1/tau }

  4. Whoah! What about R?

  5. MCMCglmm • MCMCglmm library • coda library for looking at MCMC chains

  6. The basic MCMCglmm marine_mcmc <- MCMCglmm(LR ~ 1, random = ~ Study, mev= marine$VLR, data=marine)

  7. Chain of Intercepts… (from plot)

  8. Variance Components

  9. Priors… • Specify Priors in separate list • Flat prior prior<-list(B=list(mu=c(0),V=diag(c(1e+10)))) • Informative prior prior<-list(B=list(mu=c(1),V=diag(c(2))))

  10. Summary with Credible Intervals Iterations = 3001:12991 Thinning interval = 10 Sample size = 1000 DIC: 203.4118 G-structure: ~Study post.mean l-95% CI u-95% CI eff.samp Study 0.07771 0.0101 0.1487 783.3 R-structure: ~units post.mean l-95% CI u-95% CI eff.samp units 0.1786 0.1083 0.2581 872.8 Location effects: LR ~ 1 post.mean l-95% CI u-95% CI eff.samppMCMC (Intercept) 0.17066 0.04011 0.29359 1000 0.008 ** --- Signif. codes: 0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

  11. What a Bad Chain Looks Likes(no study variances included)

  12. Multiple Chains to Diagnose Convergence marine_mcmc_1 <- MCMCglmm(LR ~ 1, random = ~ Study, mev= marine$VLR, data=marine) marine_mcmc_2 <- MCMCglmm(LR ~ 1, random = ~ Study, mev= marine$VLR, data=marine) marine_mcmc_3 <- MCMCglmm(LR ~ 1, random = ~ Study, mev= marine$VLR, data=marine)

  13. Multiple Chains to Diagnose Convergence chainList <- mcmc.list(marine_mcmc_1$Sol, marine_mcmc_2$Sol, marine_mcmc_3$Sol) chainList_vcv <- mcmc.list(marine_mcmc_1$VCV[,-2], marine_mcmc_2$VCV[,-2], marine_mcmc_3$VCV[,-2])

  14. Gelman-Rubin Diagnostics > gelman.diag(chainList) Potential scale reduction factors: Point est. Upper C.I. (Intercept) 1 1 > gelman.diag(chainList_vcv) Potential scale reduction factors: Point est. Upper C.I. Study 1 1.01 units 1 1.01 Multivariate psrf 1.01

  15. Plotting Chains Shows Convergence

  16. Plotting Chains Shows Convergence

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