Mplus Users Guide v6.. - Muthén & Muthén

CHAPTER 16BAYESIAN ESTIMATIONBayesian estimation differs from frequentist estimation in thatparameters are not considered to be constants but to be variables(Gelman et al., 2004). The parameters can be given priorscorresponding to theory or previous studies. Together with thelikelihood of the data, this gives rise to posterior distributions for theparameters. Bayesian estimation uses Markov chain Monte Carlo(MCMC) algorithms to create approximations to the posteriordistributions by iteratively making random draws in the MCMC chain.The initial draws in the MCMC chain are referred to as the burnin phase.In Mplus, the first half of each chain is discarded as being part of theburnin phase. Convergence is assessed using the Gelman-Rubinconvergence criterion based on the potential scale reduction factor foreach parameter (Gelman & Rubin, 1992; Gelman et al., 2004, pp. 296-297). With multiple chains, this is a comparison of within- and betweenchainvariation. With a single chain, the last half of the iterations is splitinto two quarters and the potential scale reduction factor is computed forthese two quarters. Convergence can also be monitored by the traceplots of the posterior draws in the chains. Auto-correlation plotsdescribe the degree of non-independence of consecutive draws. Theseplots aid in determining the quality of the mixing in the chain. For eachparameter, credibility intervals are obtained from the percentiles of itsposterior distribution. Model comparisons are aided by the DevianceInformation Criterion (DIC). Overall test of model fit is judged byPosterior Predictive Checks (PPC) where the observed data is comparedto the posterior predictive distribution. In Mplus, PPC p-values arecomputed using the likelihood-ratio chi-square statistic for continuousoutcomes and for the continuous latent response variables of categoricaloutcomes. Gelman et al. (2004, Chapter 6) and Lee (2007, Chapter 5)give overviews of model comparison and model checking. For atechnical description of the Bayesian implementation, see Asparouhovand Muthén (2010). To obtain Bayesian estimation, specify:ESTIMATOR=BAYES;PARAMETERIZATIONThe PARAMETERIZATION option is used for two purposes. The firstpurpose is to change from the default Delta parameterization to thealternative Theta parameterization when TYPE=GENERAL is used, at534