MorphZ: Enhancing evidence estimation through the Morph approximation

Abstract: We introduce the Morph approximation, a class of product approximations of probability densities that selects low-order disjoint parameter blocks by maximizing the sum of their total correlations. We use the posterior approximation via Morph as the importance distribution in optimal bridge sampling. We denote this procedure by MorphZ, which serves as a post-processing estimator of the marginal likelihood. The MorphZ estimator requires only posterior samples together with the prior and likelihood, and is fully agnostic to the choice of sampler. We evaluate MorphZ's performance across statistical benchmarks, pulsar timing array (PTA) models, compact binary coalescence (CBC) gravitational-wave (GW) simulations and the GW150914 event. Across these applications, spanning low to high dimensionalities, MorphZ yields accurate evidence at substantially reduced computational cost relative to standard approaches, and can improve these estimates even when posterior coverage is incomplete. Its bridge sampling relative error diagnostic provides conservative uncertainty estimates. Because MorphZ operates directly on posterior draws, it complements exploration-oriented samplers by enabling fast and reliable evidence estimation, while it can be seamlessly integrated into existing inference workflows.