Exact sampling of the unobserved covariates in Bayesian spline models for measurement problems

In truncated polynomial spline or B-spline models where the covariates are measured with error, a fully Bayesian approach to model fitting requires the covariates and model parameters to be sampled at every Markov chain Monte Carlo iteration. Sampling the unobserved covariates poses a major computational problem and usually Gibbs sampling is not possible. This forces the practitioner to use a Metropolis–Hastings step which might suffer from unacceptable performance due to poor mixing and might require careful tuning. In this article we show for the cases of truncated polynomial spline or B-spline models of degree equal to one, the complete conditional distribution of the covariates measured with error is available explicitly as a mixture of double-truncated normals, thereby enabling a Gibbs sampling scheme.We demonstrate via a simulation study that our technique performs favorably in terms of computational efficiency and statistical performance. Our results indicate up to 62 and 54% increase in mean integrated squared error efficiency when compared to existing alternatives while using truncated polynomial splines and B-splines respectively. Furthermore, there is evidence that the gain in efficiency increases with the measurement error variance, indicating the proposed method is a particularly valuable tool for challenging applications that present highmeasurement error.We conclude with a demonstration on a nutritional epidemiology B Anindya Bhadra [email protected] Raymond J. Carroll [email protected] 1 Department of Statistics, Purdue University, 250 N. University Street, West Lafayette, IN 47907-2066, USA 2 Department of Statistics, Texas A&M University, 3143 TAMU, College Station, TX 77843-3143, USA data set from the NIH-AARP study and by pointing out some possible extensions of the current work.