Sticky Manifold Piecewise Deterministic Monte Carlo for Bayesian Model Averaging

Assigning positive prior probability to lower-dimensional submodels enriches the range of uncertainty that Bayesian models can represent. The resulting posterior distributions combine continuous parameter uncertainty with discrete model uncertainty. Efficient movement between these submodels, however, can be difficult to design. We propose the sticky manifold construction as a principled, modular framework for combining piecewise deterministic Markov processes (PDMPs) across different submodels. The integration is rejection-free, requires no additional likelihood evaluations, and retains the non-reversible and ballistic nature of the component PDMP dynamics. We prove that the resulting process leaves the intended mixed-dimensional target invariant. We also develop a manifold PDMP algorithm based on coordinate charts to provide the within-submodel dynamics and a blow-up procedure that resolves singular sets into codimension-one sticky boundaries. This geometric construction makes the remaining design choices explicit; component dynamics and unsticking kernels can be chosen freely, subject to easily verifiable velocity compatibility conditions. Finally, we demonstrate the sticky manifold algorithms in Bayesian model averaging examples, propagating uncertainty over pure versus mixed states in quantum state tomography, rank in Bayesian principal component analysis, and parametric families in survival extrapolation.