Information-Computation Inversion in Pseudo-Marginal MCMC

Abstract

Observation refinement changes posterior uncertainty and the stochastic likelihood calculation in pseudo-marginal MCMC. We study their joint effect on finite-run posterior-functional risk. A retained-state bound localizes computational error to discrepant high-weight states. Within a common bootstrap construction, we derive an exact one-particle risk formula and a sufficient reversal condition for every fixed particle count. Finer observations can then reduce posterior uncertainty while increasing total squared-error risk. For posterior events, we allocate auxiliary computation by coupling cross-event proposals and refreshing same-event proposals independently. A swap identity establishes invariance. A continuation-risk identity describes within-event updates and joint state-cost laws. For one paired transcription record, finer observations give lower conditional total risk. Doubling the particle count raises computational MSE at a fixed CPU budget. A separate 96-state gene-network comparison gives a 21.1% reduction in event mean-squared error at a prespecified 25-second budget. A common functional-risk criterion connects observation refinement and auxiliary allocation.

90 pages, 7 figures. Includes supplementary material. Version 2 expands the theory and adds numerical experiments