Integral priors for Bayesian model selection: how they operate from simple to complex cases

In Bayesian model selection for the sake of objectivity very often default estimation priors are used. However, these priors are usually improper yielding indeterminate Bayes factors that preclude the comparison of the models. To solve this difficulty integral priors have been proposed as prior distributions for Bayesian model selection in Cano et al. (Test 17(3):493–504, 2008). These priors are the solution to a system of two integral equations, and the $$\sigma $$ σ -finite invariant measures associated with a Markov chain. They have been further developed in Cano and Salmerón (Bayesian Anal 8(2):361–380, 2013) and applied to binomial regression models in Salmerón et al. (Stat Sin 25(3):1009–1023, 2015). One of the main advantages of this methodology is that it can be applied to compare both nested and non-nested models. Here, we present some applications of this methodology along with some new technical developments, from the simplest case to more advanced ones to illustrate how it works. We begin with the toy example of a normal mean with known variance to easily point out how this methodology operates. Then, we consider the comparison of the normal location model with the double exponential one. Finally, we consider the case of integral priors for the one-way heteroscedastic ANOVA, where the simulation of the Markov chains involves a Gibbs sampling algorithm, and we present some relevant conclusions and outline oncoming research.