Keywords: Convexity, Bayesian, Minimal distance mapping, Parameter constraint, Projection operator.
In a broad variety of settings, prior information takes the form of parameter restrictions. Bayesian approaches are appealing in parameter constrained problems in allowing a probabilistic characterization of uncertainty in finite samples while providing a computational machinery for incorporation of complex constraints in hierarchical models. However, the usual Bayesian strategy of directly placing a prior measure on the constrained space, and then conducting posterior computation with Markov chain Monte Carlo algorithms is often intractable. An alternative is to initially conduct computation for an unconstrained or less constrained posterior, and then project draws from this initial posterior to the constrained space through a minimal distance mapping. This approach has been successful in monotone function estimation but has not been considered in broader settings. In this article, we develop a general theory to justify posterior projections in general spaces including for infinite-dimensional functional parameters. For tractability, we focus on the case in which the constrained parameter space corresponds to a closed, convex subset of the original space. We then provide a general formulation of the projected posterior, and show that it corresponds to a valid posterior distribution on the constrained space for particular classes of priors and likelihood functions. We also show how asymptotic properties of the unconstrained posterior are transferred to the projected posterior. Posterior projections are then illustrated through multiple examples, both in simulation studies and real data applications.