Covariance Decompositions for Accurate Computation in Bayesian Scale-Usage Models

A common modeling approach for data collected on discrete scales is to introduce continuous latent variables, treated as missing data, that are linked to the observations via a censoring mechanism. Bayesian approaches typically use data augmentation within a Markov chain Monte Carlo (MCMC) algorithm that simplifies analysis by avoiding direct evaluation of the likelihood. However, if not carefully implemented, the cost of data augmentation is that Markov chain mixing can be severely degraded, rendering the method useless for inference. Motivated by the need to consider complex, high-dimensional models where the latent variables are not independently distributed, we introduce a new covariance decomposition for the analysis of discrete data models with multivariate normal latent variables that facilitates “joint” MCMC updates of the latent variables and censoring points. Our decomposition results in a sampling algorithm that only requires univariate normal integrals, which can be evaluated with high accuracy. We provide theoretical and practical guidance for choosing a good decomposition and present an illustration that demonstrates its effectiveness over a commonly used approach that employs numerical approximations of multivariate normal integrals over rectangular regions, which, when repeated many times in an MCMC algorithm, can have a substantial impact on the chain’s limiting distribution.