Expectations under a product of densities now need no density: one regression merges component samples, error 0.0001 on HIGGS

Expectations under a product distribution can now be computed from component samples alone, without writing down or evaluating the component densities. Earlier routes either approximated each component with a parametric family, reweighted samples by importance weights, or built a fusion sampler targeting the product, all of which rely on density evaluations or parametric assumptions. On 11 million simulated HIGGS collision events, PMC-MLP reaches an estimation error of 0.0001 with 0.1092 minutes of merge time and about 10% of full-data MCMC total runtime; GBF is computationally infeasible at this scale and omitted.

The construction writes the product distribution as the conditional law of a consensus given that a discrepancy is zero, then regresses on the discrepancy and evaluates at zero. The consensus summarizes the components and the discrepancy measures their disagreement; the conditional mean is estimated by regression rather than by density ratios or particle fusion. On simulated cryo-EM images of the HSP90 protein, each image first yields a subposterior through neural posterior estimation and PMC then merges them; PMC-MLP nearly exactly recovers the full-data posterior mean and PMC-GP is very close, while sampling each image-level subposterior with SBI alone takes 1.7 minutes and PMC's additional cost is negligible.

The method targets researchers and engineers who must merge several independently computed inferences, under the condition that only samples from each component are available and densities cannot be evaluated. Demonstrated settings include simulated data, a privacy-restricted respiratory clinical trial, large-scale HIGGS inference, and simulation-based inference on HSP90 protein images. When component overlap is limited, the local effective sample size falls and estimation relies more on extrapolation; larger posterior uncertainty or failed posterior predictive checks flag this, and sampling the full target directly or redesigning components for better overlap is the stated recourse.

The convergence-rate analysis assumes independent and identically distributed component samplers, while consistency for MCMC sampling holds under standard ergodicity or mixing conditions. Limited component overlap reduces the local effective sample size and makes aggregation rely more on extrapolation, which the posterior uncertainty reflects. Large-scale inference carries a tradeoff: smaller batches lower local sampling cost but can yield more divergent subposteriors, while larger batches improve overlap at greater sampling cost.