Geometric ergodicity of Gibbs samplers for Bayesian error-in-variable regression

We consider Bayesian error-in-variable (EIV) linear regression accounting for additional additive Gaussian error in the features and response. We construct 3-variable deterministic scan Gibbs samplers for EIV regression models using classical and Berkson errors with independent normal and inverse-gamma priors. We prove these Gibbs samplers are always geometrically ergodic which ensures a central limit theorem for many time averages from the Markov chains.