MCMC with Measurement Error

You need some basis for quantifying error variance — ideally a validation sub-study with both error-prone and gold-standard measurements, replicate measures whose variance can be modelled, or an informative prior from published assay reliability data. If no such information exists, the model is not identifiable and the correction cannot be trusted.

Regression calibration is a simpler frequentist approximation that replaces the error-prone covariate with its conditional expectation given the observed value. MCMC with measurement error is fully Bayesian: it samples the latent true values and propagates all sources of uncertainty — including uncertainty about the error variance itself — through the posterior, making it more accurate in nonlinear models and small samples.

Yes. Replace the additive error model with a misclassification matrix (the probability of observing each category given the true category). With a prior on that matrix or data from a validation study, Gibbs sampling can impute the true category for each subject alongside the structural parameters.

In a simple linear model with a single predictor, classical additive error does attenuate coefficients toward zero. In multivariate models with multiple error-prone covariates the bias pattern is more complex — coefficients can be attenuated or even inflated — which is another reason to model measurement error explicitly rather than ignore it.

Run multiple chains from dispersed starting values and monitor R-hat for each parameter including the latent X values. Values below about 1.01 and sufficient effective sample sizes indicate convergence. Also inspect trace plots; the added latent variables increase dimensionality, so longer chains or stronger sampler adaptation (e.g. NUTS in Stan) are often needed.