Stanmanual on measurement error modelling does not seem to recommend, as a default, including a so-called
exposure model. Should this be the case? Say I have imperfectly measured surrogate x^* and error free covariates z. Assume a strong correlation between the two. The Bayesian measurement error literature factors the joint distribution of perfectly measured variables into: f(x,z) = f(x|z) f(z) —> where f(x|z) is the so-called exposure model. But with a correlation between x^* and z - which is usually the case when working with an ME model (!) - should this exposure model be optional? Since if one instead uses f(x) – instead of f(x|z) – is one not implicitly assuming independence between x and z?
Why does the Stan manual not put a prior on \tau, the variance of the measurement error model? As an unknown parameter, in a Bayesian approach I can’t see why it should not have a prior. The measurement error textbooks seem to include a prior on \tau (e.g. the books by Gustafson, 2004; Caroll et al, 2006; Grace Y Yi; 2017).
In our case, we are running a ‘sensitivity analysis’, putting in different values for \tau and seeing how the results change. But even then, should we have a prior for \tau?