On the use of pseudo-likelihoods in Bayesian variable selection.

In the presence of nuisance parameters, we discuss a one-parameter Bayesian analysis based on a pseudo-likelihood assuming a default prior distribution for the parameter of interest only. Although this way to proceed cannot always be considered as orthodox in the Bayesian perspective, it is of interest to evaluate whether the use of suitable pseudo-likelihoods may be proposed for Bayesian inference. Attention is focused in the context of regression models, in particular on inference about a scalar regression coefficient in various multiple regression models, i.e. scale and regression models with non-normal errors, non-linear normal heteroscedastic regression models, and log-linear models for count data with overdispersion. Some interesting conclusions emerge.