Information-Computation Inversion in Pseudo-Marginal MCMC
Pseudo-marginal MCMC is exact under nonnegative unbiased likelihood estimation, but observation design can change both posterior information and the stochastic law of the likelihood estimator. We study fixed-horizon recovery of a declared posterior functional under this joint change. For a broad fresh-estimator pseudo-marginal class, we derive an inverse-weight acceptance bound that yields a lower bound on functional mean-squared error from high retained-weight mass in discrepant states. A two-region corollary establishes an information-computation inversion: stronger exact posterior separation can coexist with worse finite-horizon recovery when the estimator law changes. The obstruction motivates an exact route-conditioned transition that allocates coupled auxiliary inheritance and independent refreshment according to a scientific pair route. In a controlled transcription experiment, a finer observation strengthens likelihood separation while degrading particle-filter reliability and finite-chain behavior. In a fresh 96-state gene-network comparison, route conditioning reduces measured particle-filter wall time by 42.4% in the frozen execution (95% retained-state bootstrap interval 39.6%-45.2%). Recovery uncertainty spans the predeclared noninferiority margin. A prospective Lotka-Volterra experiment supports portability of the retained-state route-restoration mechanism. The results distinguish statistical information, retained-state functional risk, and auxiliary-computation allocation in pseudo-marginal inference.