A note on the relationships between Bayesian and non-Bayesian predictive inference

Let us consider a random vector Y distributed according to a statistical model depending on an unknown parameter vector. Suppose we are interested in prediction, where the object of inference is a future or a yet unobserved random variable X, not depending on Y. The aim of this contribution is to provide some theoretical insight into the relationships between Bayesian and non-Bayesian higher-order asymptotic expansions for predictive densities. More precisely, we characterize prior probability distributionswhich guarantee the equivalence between Bayesian asymptotic expansions of the predictive distribution and frequentist accurate refinements of the estimative predictive density. An illustration in the context of the scalar skew-normal model is discussed. As a further result, invoking asymptotic connections between adjustments of the profile likelihood and asymptotic expansions of predictive densities, we illustrate how Bayesian predictive densities allows for the construction of adjusted profile loglikelihoods.