The matryoshka doll prior: principled multiplicity correction in Bayesian model comparison

This paper introduces a general and principled construction of model space priors with a focus on regression problems. The proposed formulation regards each model as a `local` null hypothesis whose alternatives are the set of models that nest it. Assuming constant odds between any `local` null and its alternatives provides a natural isomorphism of model spaces (like a matryoshka doll), constituting an intuitive way to correct for test multiplicity. This isomorphism yields the Poisson distribution as the unique limiting distribution over model dimension under mild assumptions. We compare this model space prior theoretically and in simulations to widely adopted Beta-Binomial constructions. We show that the proposed prior yields a `just-right` multiplicity correction that induces a desirable complexity penalization profile.