HAPERA: Residual Allocation for Amortized Posterior Estimation in Hierarchical Populations

Applications from device fleets to patient cohorts often require inferring the global parameters of a hierarchical model from short early observation windows. Amortized simulation-based inference trains a network once on simulated populations and reuses it without retraining. Because populations evolve at very different speeds across the prior, simulators may set windows to a random fraction of the population lifetime implied by the global parameters. We show that an ideally trained estimator then treats the window length as evidence about these parameters, even though their marginal prior is unchanged. At deployment, where windows are chosen independently of the parameters, this evidence is spurious, so the learned posterior is biased. Simulation-based calibration (SBC) on the training simulations cannot reveal this bias, because the ideal estimator is calibrated for those simulations. We propose HAPERA, a flow-matching estimator of the joint posterior over global and unit-level parameters. First, it trains on windows drawn from their marginal distribution, independently of the parameters. Second, it adds an allocation residual, a vector-field term dedicated to shifting the population location and every unit's deviation from it equally in opposite directions. Such a shift leaves each unit's parameter, the location plus its deviation, and hence the likelihood, unchanged, so only the prior can tell these allocations apart. On a synthetic benchmark, a classifier two-sample test places HAPERA's posterior samples clearly nearer NUTS reference samples than those of the hierarchical amortized estimators HNPE and TFMPE, trained under the same window design, at every window length and seed. On two real-world datasets, LED degradation and drug pharmacokinetics, HAPERA is nearly indistinguishable from the NUTS reference and closer than both estimators. In ablations, parameter-dependent windows pull the posterior away from the reference, most for the shortest windows. Removing the residual also pulls it away at every training seed.