A structural restriction in a statistical model adds valuable information and is often necessary for a proper scientific interpretation. However, incorporating it in Bayesian analysis makes the posterior distribution substantially more complex for computation and analyzing its behavior. We propose an extension of the Bayesian paradigm where a structure-complying correction map from an unrestricted posterior induces the structural restriction. The resulting "immersion posterior" offers a simple and effective solution to address structural restrictions. We give Bayesian interpretations of the immersion posterior and compare it with other extensions of the Bayesian paradigm. We illustrate how the immersion posterior can effectively achieve optimal posterior concentration and frequentist coverage of Bayesian credible regions in various statistical models with structural restrictions.
