Comparison between a measurement error model and a linear model without measurement error

The regression of a response variable y on an explanatory variable ξ from observations on (y,x), where x is a measurement of ξ, is a special case of errors-in-variables model or measurement error model (MEM). In this work we attempt to answer the following question: given the data (y,x) under a MEM, is it possible to not consider the measurement error on the covariable ξ in order to use a simpler model? To the best of our knowledge, this problem has not been treated in the Bayesian literature. To answer that question, we compute Bayes factors, the deviance information criterion and the posterior mean of the logarithmic discrepancy. We apply these Bayesian model comparison criteria to two real data sets obtaining interesting results. We conclude that, in order to simplify the MEM, model comparison criteria can be useful to compare structural MEM and a random effect model, but we would also need other statistic tools and take into account the final goal of the model.