Information-Computation Inversion in Pseudo-Marginal MCMC
Observation refinement changes posterior uncertainty and the likelihood calculation in pseudo-marginal MCMC. We compare their combined effect through finite-run squared-error risk. A sufficient inversion condition relates information gain to accepted event flow and coarse-kernel contraction. A bootstrap construction realizes inversion at every fixed particle count. We then couple cross-event proposals and refresh same-event proposals independently. A swap identity establishes invariance; continuation identities describe subsequent risk. In the same finite model, a rational certificate proves inversion against optimized constant mixtures and repair by selective allocation over an initialization class at a common action-price budget. Reaction-network experiments measure CPU costs. Under finite-pool initialization, selective allocation reduces event mean-squared error by 40.5% against a tuned mixture at 25 post-initialization CPU seconds. Paired transcription observations show how increased particle effort can raise finite-budget error.