Subsampled Pseudo-posteriors for Scalable Bayesian Moment-condition Inference
Bayesian pseudo-posteriors based on moment conditions, such as Bayesian empirical likelihood (EL) and Bayesian exponentially tilted empirical likelihood (ETEL), provide a robust route to Bayesian inference when the model is specified only through moment restrictions, but their computation is often prohibitive. In this work, we address two barriers to Bayesian pseudo-inference under moment restrictions. First, scalable sampling is challenging for tall data because the pseudo-likelihoods are not log-additive across observations, so standard likelihood-subsampling methods do not apply. Second, in simulation-based and latent-variable problems, the moment function itself may be defined through an intractable expectation, making full-data pseudo-posterior inference infeasible. We develop a new family of subsampled pseudo-posteriors that replaces the full-data pseudo-posterior with an aggregate of mini-batch pseudo-posteriors. We show that naive mini-batching induces a shifted-mixture distortion, yielding aggregates that are correctly centered but overly diffuse. To remove this distortion, we introduce moment-function-level control variates that align mini-batch moment equations around a reference estimator and recover the full-data posterior shape. For Bayesian ETEL, we establish finite-sample total variation bounds to the full-data pseudo-posterior, allowing discontinuous and intractable moment functions. Numerical experiments show that the method applies broadly across moment-condition-based Bayesian pseudo-inference, closely approximates full-data pseudo-posteriors, and substantially reduces computational cost.