Bayesian estimation and testing of structural equation models

The Gibbs sampler can be used to obtain samples of arbitrary size from the posterior distribution over the parameters of a structural equation model ( SEM) given covariance data and a prior distribution over the parameters. Po int estimates, standard deviations and interval estimates for the parameter s can be computed from these samples. If the prior distribution over the pa rameters is uninformative, the posterior is proportional to the likelihood, and asymptotically the inferences based on the Gibbs sample are the same a s those based on the maximum likelihood solution, for example, output from LISREL or EQS. In small samples, however, the likelihood surface is not Gau ssian and in some cases contains local maxima. Nevertheless, the Gibbs samp le comes from the correct posterior distribution over the parameters regard less of the sample size and the shape of the likelihood surface. With an in formative prior distribution over the parameters, the posterior can be used to make inferences about the parameters of underidentified models, as we i llustrate on a simple errors-in-variables model.