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Product Monte Carlo: Expectation Under a Product of Densities

Pietro Carlotti, Sifan Liu, Yuling Yao

arXiv:2610.12702v1stat.MEcs.LGstat.CO

TL;DR

Computing expectations under a product of component densities is challenging when only separate component samples are available. PMC expresses the target through a conditional expectation and estimates it with regression, with a Bayesian formulation for uncertainty quantification and diagnostics. The paper reports improved accuracy and lower computational cost than benchmarks in its experiments, alongside theoretical Gaussian efficiency.

  • Problem

    Computing expectations under a product density remains nontrivial when component densities cannot be evaluated or when component samples are obtained separately.

    Source: §1 · pp. 2, 4

  • Method

    PMC uses regression on consensus and discrepancy variables to estimate the conditional expectation at zero discrepancy from component samples.

    Source: §1 · pp. 2–3

  • Results

    PMC-OLS closely matches the Oracle in MSE, and PMC methods outperform GBF at moderately large sample sizes while often running orders of magnitude faster.

    Source: §4 · p. 11

  • Takeaways & Limitations

    PMC is density-evaluation-free and distribution-free, and its Bayesian formulation provides uncertainty quantification and supports model diagnostics.

    Source: §1 · p. 3, §4 · p. 11

    Limited component overlap reduces local effective sample size and increases reliance on extrapolation; the authors recommend full-target sampling when feasible or redesigning components to improve overlap.

    Source: §5 · p. 13

Abstract

from arXiv · show

Aggregating independently computed sub-sample inferences is a well-known challenge in large-scale Bayesian computation, federated learning, and opinion pooling. The core problem is to approximate a distribution whose density is proportional to a product of individual densities. We propose Product Monte Carlo (PMC), an algorithm for computing arbitrary expectations under a product of densities using only Monte Carlo draws from individual densities. We turn the intractable product expectation into a regression problem by expressing it through a family of computable conditional expectations. Besides the point estimate, our fully Bayesian regression formulation provides coherent uncertainty quantification and principled diagnostics. We establish theoretical guarantees on consistency and characterize convergence rates for the PMC estimator. Across simulated and real-data experiments varying the number, dimensionality, and heterogeneity of the component distributions, PMC achieves higher estimation accuracy, better-calibrated uncertainty coverage, and improved scalability at a lower computational cost than existing methods. In addition, PMC requires no density evaluation, making it applicable to simulation-based inference, as we demonstrate in an application with protein images.

1. Introduction

The paper targets expectations under pointwise products of component densities, a task that remains difficult even when component samples can be obtained separately. PMC recasts the target as a regression problem and uses Bayesian modeling to quantify uncertainty and diagnose the fit.

1. Introduction

  • Computing full-data posterior quantities remains nontrivial because the posterior is proportional to the product of subposteriors.The paper defines the product as a pointwise density product, not a product measure.Source: §1 · p. 2
  • PMC estimates arbitrary expectations under the product distribution by aggregating component samples through regression, without evaluating component densities.The conditional representation expresses the expectation through consensus given zero discrepancy, estimated by regression.Source: §1 · pp. 2–3
  • Product densities arise in distributed inference, privacy-sensitive settings, simulation-based inference, and predictive pooling.Component densities may be unavailable or intractable to evaluate in these settings.Source: §1 · pp. 3–4

1.1. Related methods

  • PMC makes no parametric assumptions and recovers parametric efficiency in the Gaussian case, whereas parametric approximations can retain bias even with large Monte Carlo samples.The related-methods discussion identifies this efficiency as a theoretical result.Source: §1.1 · pp. 4–5
  • Importance weighting can be unstable with heavy-tailed ratios, while fusion methods require density and derivative evaluations and incur substantial simulation overhead.These constraints can limit applicability when densities are unavailable or costly to evaluate.Source: §1.1 · p. 5
  • PMC’s Bayesian formulation reflects epistemic as well as Monte Carlo uncertainty, including difficulty when component distributions are well-separated.The paper contrasts this with methods whose uncertainty can vanish as Monte Carlo size grows, even for inaccurate estimates.Source: §1.1 · p. 5

2. Proposed method

PMC estimates expectations under a product distribution from independent component samples by relating the target to a conditional expectation. It fits that conditional relationship with regression and can use a Bayesian formulation for uncertainty quantification and diagnostics.

2.1. Consensus and discrepancy

  • The product distribution is represented as the consensus mapping’s law conditional on discrepancy being zero.The conditional consensus law for discrepancies near zero converges in total variation to the product law, under the stated integrability assumption.Source: §2.1 · p. 6
  • A consensus mapping returns the common value for identical inputs and is translation-equivariant; discrepancy mappings are designed so zero implies all component values agree.Examples of consensus include the arithmetic average, weighted average, and median.Source: §2.1 · p. 6

2.2. Conditional mean via regression

  • PMC regresses h(C) on D to estimate E[h(C)|D], then evaluates the fitted conditional mean at zero discrepancy.A flexible neural network can provide a point estimate by minimizing empirical squared error.Source: §2.2 · p. 7

2.3. Fully Bayesian formulation, uncertainty quantification, and diagnostics

  • The fully Bayesian formulation places a Gaussian-process prior on the conditional mean and returns posterior uncertainty for the target expectation.Posterior predictive checks can flag an inadequate regression specification, such as a linear model when C given D is nonlinear.Source: §2.3 · pp. 7–8
  • PMC’s posterior variance reflects Monte Carlo and epistemic uncertainty, with data determining how observations away from zero discrepancy inform the target.In the stated GP setting, the variance can remain τ^2 as S grows when the lengthscale tends to zero.Source: §2.3 · p. 8

3. Asymptotic theory

PMC identifies the target product expectation with a conditional mean at zero, making consistency follow from consistent estimation of that regression target. Its convergence behavior depends on discrepancy density, regression properties, conditional variance, and discrepancy dimension; in the Gaussian case, it can attain parametric efficiency.

Consistency

  • A consistent estimator of E[h(C) | D = 0] consistently estimates the target product expectation; this requires independence across components, not i.i.d. draws within samplers.PMC also supports MCMC sampling, retaining consistency under standard ergodicity or mixing conditions.Source: §3 · p. 8

Convergence rate

  • The convergence-rate characterization depends on discrepancy density at zero, conditional-mean curvature, conditional variance, and discrepancy dimension.The local-linear rate analysis assumes i.i.d. component samples to simplify the theory.Source: §3 · pp. 8–9

When PMC is harder

  • Lower discrepancy density at zero means fewer samples near consensus and greater reliance on extrapolation; it decreases as components move farther apart or become more diffuse.The discrepancy density at zero determines the local effective sample size; the proposition relates it to generalized Cauchy–Schwarz divergence and entropy.Source: §3 · p. 9

Gaussian example

  • PMC’s leading-order AMSE for the Gaussian product mean matches the Cramér–Rao lower bound, attaining the efficiency of the closed-form Gaussian estimator using samples alone.The result concerns h(u) = u; consistency holds for any admissible representation, but scalar discrepancy attains the bound only with precision weighting.Source: §3 · p. 10
  • In the Gaussian example, the choice of consensus and discrepancy representation affects convergence rate; scalar discrepancy reaches the Cramér–Rao bound only with precision weighting.With d = δ, any linear consensus attains the same rate; the experiments use arithmetic-average consensus and full discrepancy.Source: §3 · p. 10

4. Experiments

Experiments compare PMC with fusion methods and full-data inference across simulations and real applications. PMC shows favorable accuracy, computational cost, and uncertainty behavior, including in likelihood-free protein imaging.

4. Experiments

  • Benchmarks vary component family, separation, sample size, dimension, and component count, assessing expectation-estimation MSE and wall-clock training time.The simulated benchmarks include a parametric Oracle; PMC is the only method considered that is both density-evaluation-free and distribution-free.Source: §4 · p. 11
  • At moderately large sample sizes, all PMC methods outperform GBF in accuracy and are substantially faster, often by orders of magnitude.PMC-OLS closely matches the Oracle in MSE and improves with S; GBF results are shown only where computationally feasible.Source: §4 · p. 11, Figure 4 · p. 12
  • PMC-GP uncertainty accounts for Monte Carlo and epistemic uncertainty; its coverage is reasonable, while GBF intervals become increasingly undercovered as particle count rises.For two Gaussian components, PMC-GP posterior uncertainty increases as their means separate, warning of limited overlap and a harder product-estimation problem.Source: §4 · p. 11, Figure 5 · p. 13, Figure 6 · p. 13
  • The privacy-restricted clinical-trial experiment aggregates clinic subposteriors and compares estimates with pooled-data MCMC, separating local sampling cost from aggregation cost.The pooled-data MCMC serves as an oracle truth despite the assumed privacy constraint.Source: §4 · p. 11, Table 2 · p. 12
  • On HIGGS, PMC achieves small estimation errors at roughly 10% of full-data MCMC runtime; GBF is computationally infeasible at this scale.The experiment uses 11 million simulated collision events split into 10 batches.Source: §4 · p. 12, Table 3 · p. 13

Simulation-based inference

  • PMC-MLP nearly exactly recovers the full-data posterior mean for protein imaging, PMC-GP is very close, and PMC's added cost is negligible.Sampling each image-level subposterior with SBI takes 1.7 minutes.Source: Simulation-based inference · pp. 12–13

5. Discussion

Limited overlap reduces local effective sample size and increases reliance on extrapolation. PMC's uncertainty and predictive checks can warn of unreliable estimates; the discussion recommends improving overlap or sampling from the full target when feasible.

5. Discussion

  • Limited component overlap reduces local effective sample size, making aggregation more reliant on extrapolation.Large posterior uncertainty or failed posterior predictive checks warn that samples may not support reliable estimation.Source: §5 · p. 13
  • When overlap is poor, the authors recommend sampling from the full target when feasible or redesigning components to improve overlap.Smaller batches lower local sampling cost but can yield more divergent subposteriors; larger batches improve overlap at greater sampling cost.Source: §5 · p. 13
  • Future directions include adapting regression choices, incorporating target knowledge, and extending conditional-mean regression to conditional-density estimation for generative sampling.The current implementation uses fixed choices of c and d.Source: §5 · pp. 13–14

Product Monte Carlo: Supplementary Materials

The supplementary materials develop coordinate-based proofs, special cases, and local linear regression for PMC, and establish consistency and bandwidth results while deriving a Jacobian formula for the constant in Proposition 1.

A.1. Proof of Theorem 1

  • Under Theorem 1’s assumptions, conditioning on sufficiently small discrepancy is well-defined, and the consensus law converges in total variation to the product law.The proof establishes positivity of the conditioning event and a well-defined conditional density before showing convergence.Source: §A.1 · pp. 20–21

A.2. Weighted-Average Consensus

  • The weighted-average consensus uses matrices whose sum is the identity and is continuous, reproduces diagonal inputs, and is translation-equivariant.The accompanying result uses the full discrepancy mapping δ, with ψ = id and range dimension q = (K − 1)p.Source: §A.2 · p. 21

A.3. Proof of Proposition 3

  • With weighted-average consensus and the full discrepancy map, the conditional law of the consensus given null discrepancy is the product law.The proof identifies the product density with the conditional density at zero discrepancy.Source: §A.3 · p. 22

A.4. Gaussian Components

  • For Gaussian components, an appropriate consensus can marginally recover the product distribution and be independent of discrepancy.The stated proposition assumes Gaussian component distributions with positive-definite covariance matrices.Source: §A.4 · p. 23

A.5. Proof of Proposition 4

  • The Gaussian proof derives independence because consensus and discrepancy are jointly Gaussian and uncorrelated.It also establishes that the consensus is distributed according to the product distribution.Source: §A.5 · p. 24

A.6. Proof of Theorem 2

  • Theorem 2’s proof introduces local linear regression as a kernel-and-bandwidth optimization estimator of a conditional mean from i.i.d. data.The excerpt begins stating regularity assumptions concerning the density of D but does not provide the theorem’s result.Source: §A.6 · pp. 24–25
  • Under Assumptions A.1–A.5, the proof establishes estimator consistency and derives the leading-order AMSE and its minimizing bandwidth.The proof uses standard local linear regression expansions.Source: §A.6 · pp. 25, 27

A.7. Proof of Proposition 1

  • The proof of Proposition 1 identifies Cψ as 1/|det(Jψ(0q))|.This formula follows from a change of variables under the stated regularity assumptions.Source: §A.7 · p. 29

A.8. Proof of the Entropy Lower Bound in Proposition 1

The appendix proves the entropy lower bound and analyzes the Gaussian case for Proposition 2. It also outlines the supplementary experimental results reported in later sections.

A.8. Proof of the Entropy Lower Bound in Proposition 1

  • Jensen’s inequality and an entropy-based lower bound on each component’s L_K norm are combined with Proposition 1 to prove the entropy lower bound.The proof first obtains a lower bound on the product of component L_K norms.Source: §A.8 · p. 30

A.9. Proof of Proposition 2

  • For Gaussian components, the local linear estimator is unbiased when its weighted design matrix is invertible, and the proof derives its asymptotic MSE and consistency.The analysis considers both full-vector and scalar discrepancy choices.Source: §A.9 · pp. 31–33

B. Additional Experimental Results

B.1. Additional Simulation Results

These supplementary simulations vary component count, location, scale, variance balance, and correlation in Gaussian settings. PMC methods are generally faster than GBF, while their relative accuracy depends on the target, sample size, and configuration.

B.1. Additional Simulation Results

  • The supplementary simulations vary component family, component distances, sample size, dimension, and number of components, using closed-form product families and an Oracle benchmark.Source: §B.1 · p. 34

B.1.1. Univariate Gaussian: Location and Scale

  • Across location and scale settings, PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u2; all PMC methods train considerably faster than GBF.PMC-MLP and PMC-GP overtake GBF as sample size increases for h(u) = u, while for h(u) = u2 only PMC-GP overtakes it in the shown range.Source: §B.1.1 · p. 35, Figure 8 · p. 35, Figure 9 · p. 35

B.1.2. Univariate Gaussian: Number of Components

  • For K ≤ 5, PMC-MLP and PMC-GP overtake GBF at S ≥ 5 × 10^5 for h(u) = u; increasing K leaves their runtime largely unaffected while GBF runtime rises.PMC-OLS closely matches the Oracle for h(u) = u but fails for h(u) = u2; all PMC methods are considerably faster than GBF, with PMC-OLS fastest.Source: Figure 10 · p. 36, Figure 11 · p. 36, §B.1.2 · p. 37

B.1.3. Bivariate Gaussian: Mean Separation

  • PMC-OLS closely matches the Oracle, while PMC-MLP and PMC-GP trail GBF by about 2 to 5 in MSE through S = 10^5; at S = 10^6 they match or improve on GBF only at smaller separations.All PMC methods train considerably faster than GBF, with PMC-OLS fastest.Source: §B.1.3 · p. 38, Figure 14 · p. 38, Figure 15 · p. 38

B.1.4. Bivariate Gaussian: Isotropic Variance Scale

  • As the second component’s isotropic scale increases, PMC-MLP estimation becomes harder; PMC-GP overtakes GBF at S ≥ 5 × 10^5, while PMC-MLP reaches parity at S = 10^6.PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u ⊙u; all PMC methods train considerably faster than GBF.Source: §B.1.4 · p. 39, Figure 16 · p. 39, Figure 17 · p. 39

B.1.5. Bivariate Gaussian: Marginal Variance Imbalance

  • With marginal variance imbalance, PMC-GP overtakes GBF at S ≥ 5 × 10^5, while PMC-MLP reaches parity for h(u) = u and remains behind for h(u) = u ⊙u.PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u ⊙u; PMC methods train considerably faster than GBF.Source: §B.1.5 · p. 40, Figure 18 · p. 40, Figure 19 · p. 40

B.1.6. Bivariate Gaussian: Correlation Strength and Sign

  • As sample size increases across correlation settings, PMC-MLP and PMC-GP overtake GBF in most settings; PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u ⊙u.All PMC methods train considerably faster than GBF, with PMC-OLS fastest.Source: §B.1.6 · p. 41, Figure 20 · p. 41, Figure 21 · p. 41

B.1.7. Multivariate Gaussian: Dimension

The final simulations examine multivariate Gaussian dimension and symmetric bimodal Gaussian mixtures. Results show that relative accuracy and runtime vary with dimension and target function.

B.1.7. Multivariate Gaussian: Dimension

  • At p = 20, PMC-MLP and PMC-GP outperform GBF, whereas at p ≤ 5 they lag; PMC-MLP matches GBF at p = 10 for large sample sizes.PMC-OLS closely matches the Oracle for h(u) = u; GBF runtime scales poorly with dimension, while PMC methods remain far faster.Source: §B.1.7 · p. 42, Figure 22 · p. 42

B.1.8. Gaussian Mixture: Symmetry

  • For symmetric bimodal mixtures, PMC-MLP and PMC-GP outperform GBF for h(u) = u at almost all sample sizes but underperform it for h(u) = u2.PMC-OLS has lower MSE than the Oracle for h(u) = u but fails for h(u) = u2; all PMC methods train considerably faster than GBF.Source: §B.1.8 · p. 43, Figure 24 · p. 43, Figure 25 · p. 43

B.1.9. Gaussian Mixture: Mixture Weights

The appendix tests PMC on Gaussian-mixture products with opposing mode weights and heterogeneous scales. Results vary by target moment and regression specification, while PMC methods train faster than GBF in both settings.

B.1.9. Gaussian Mixture: Mixture Weights

  • PMC-OLS beats the Oracle for h(u) = u but fails for h(u) = u2; PMC-MLP and PMC-GP beat GBF for h(u) = u but lag for h(u) = u2.All PMC methods train considerably faster than GBF, with PMC-OLS fastest, even faster than the Oracle.Source: §B.1.9 · p. 44, Figure 26 · p. 44, Figure 27 · p. 44

B.1.10. Gaussian Mixture: Mixture Scales

  • PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u2; PMC-MLP and PMC-GP lag GBF at small sample sizes and are comparable at intermediate sizes.All PMC methods train considerably faster than GBF, and PMC-OLS is fastest, even faster than the Oracle.Source: §B.1.10 · p. 45, Figure 28 · p. 45, Figure 29 · p. 45

B.1.11. Gaussian Mixture: Number of Components

This experiment examines products of three Gaussian-mixture components, whose product has 12 terms. Results show that PMC performance depends on model specification and target moment, while its methods train faster than GBF.

B.1.11. Gaussian Mixture: Number of Components

  • The product of three Gaussian-mixture components with differing modes, weights, and scales is a 12-term Gaussian mixture.The experiments evaluate h(u) = u and h(u) = u2.Source: §B.1.11 · pp. 45–46
  • PMC-MLP underperforms GBF for S ≤ 5 × 10^4 but outperforms it for h(u) = u at S = 10^5; PMC-GP underperforms GBF at every tested sample size.PMC-OLS fails to estimate both moments under this misspecified setting; all PMC methods train considerably faster than GBF.Source: §B.1.11 · p. 46, Figure 30 · p. 46, Figure 31 · p. 46

B.1.12. Gaussian Mixture: Heterogeneity

Across heterogeneous Gaussian mixtures and Beta products with varied concentration, skewness, and boundary behavior, PMC results depend on model specification and target moment. The experiments also report PMC training times as considerably lower than GBF’s.

B.1.12. Gaussian Mixture: Heterogeneity

  • For heterogeneous Gaussian mixtures, PMC-OLS fails on both moments; PMC-MLP and PMC-GP beat GBF for h(u) = u, while for h(u) = u2 PMC-MLP is comparable and PMC-GP underperforms.All PMC methods train considerably faster than GBF, with PMC-OLS fastest.Source: Figure 32 · p. 47, Figure 33 · p. 47, §B.1.12 · p. 48

B.1.13. Beta: Concentration

  • For symmetric Beta products, PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u2; PMC-MLP and PMC-GP overtake GBF for h(u) = u2 at S ≥ 5 × 10^5 outside the uniform case.For h(u) = u, PMC-MLP and PMC-GP outperform GBF at large sample sizes in both concentration settings; PMC methods train faster than GBF.Source: §B.1.13 · p. 48, Figure 34 · p. 48, Figure 35 · p. 49

B.1.14. Beta: Skewness

  • For skewed Beta products, PMC-MLP and PMC-GP underperform GBF in the matching case, but are comparable at S = 10^5 and tend to outperform it at larger sizes in the opposing case.PMC-OLS fails on both moments in the matching case; in the opposing case it matches the Oracle for h(u) = u but fails for h(u) = u2. PMC methods train faster than GBF.Source: §B.1.14 · p. 50, Figure 38 · p. 50, Figure 39 · p. 50

B.1.15. Beta: Boundary Singularities

  • With Beta components diverging at opposite unit-interval endpoints, PMC-OLS matches the Oracle for h(u) = u but fails for h(u) = u2; PMC-MLP and PMC-GP beat GBF only at S = 10^6.The product distribution is bounded despite the unbounded component densities; all PMC methods train faster than GBF.Source: §B.1.15 · p. 51, Figure 42 · p. 51, Figure 43 · p. 51

B.2. Additional Real-Data Results

The MAGIC results favor PMC-GP on estimation error with negligible merging cost, while the fusion-baseline comparisons find GBF at least as accurate and fast as MCF and BF. Supplementary sections detail experimental settings and computational constraints.

B.2. Additional Real-Data Results

  • On MAGIC, PMC-GP has the lowest estimation error among merging methods, followed closely by PMC-MLP, then GBF and PMC-OLS.GBF merging takes about 27 minutes versus less than one minute for batch MCMC, making its total runtime about 28 times full-data MCMC; PMC merging is negligible and total runtimes are comparable to full-data MCMC.Source: §B.2 · p. 52, Table 4 · p. 52

B.3. Fusion Baseline Comparison

  • Across the fusion comparison, GBF is at least as accurate and as fast as MCF and BF, with its advantage growing with dimension.The individual settings qualify this overall comparison: MCF and GBF are comparable in univariate and bivariate accuracy, while GBF is more accurate in the multivariate settings.Source: §B.3 · p. 52

B.3.1. Univariate Gaussian: Location and Scale

  • For univariate Gaussian products, MCF and GBF have comparable MSE and runtime, while BF is decisively less accurate and slowest; BF runtime increases with sample size.MCF and GBF runtime is almost constant across the tested settings.Source: §B.3.1 · p. 54

B.3.2. Bivariate Gaussian: Mean Separation

  • For bivariate Gaussian products, MCF and GBF have comparable MSE, while BF is decisively worse; GBF is fastest, followed by MCF and BF.MCF runtime increases with mean separation, whereas GBF runtime grows more slowly; BF runtime increases with sample size.Source: §B.3.2 · p. 55, Figure 46 · p. 55, Figure 47 · p. 55

B.3.3. Multivariate Gaussian: Dimension

  • For multivariate Gaussian products, GBF is more accurate than MCF and BF is decisively worse; runtime ranks GBF fastest, then MCF, then BF.MCF runtime increases with dimension while GBF runtime grows more slowly; BF runtime increases with sample size.Source: §B.3.3 · p. 56, Figure 48 · p. 56, Figure 49 · p. 56

B.4. Computational Setup

  • The HSP90 simulator cannot be redistributed, so the repository alone cannot reproduce the complete pipeline or rerun simulator-dependent stages.Saved posterior draws, reported outputs, and downstream analysis code are provided; regenerating images requires access to the external simulator.Source: §B.4 · p. 57

C. Hyperparameters; C.1. Shared Baseline Simulation Settings

  • The shared baseline suite uses PMC-OLS, PMC-MLP, PMC-GP, GBF, and the Oracle, with two target functions, 20 replications, and mostly K = 2 components.Sample sizes range from 100 to 10^6, with 2 × 10^6 used only in the univariate Gaussian location-and-scale experiment.Source: C. Hyperparameters · p. 57, §C.1 · p. 58

C.2. Common PMC Settings; C.3. Deviations from the Common Settings

  • Exact GP inference costs O(S^3) time and O(S^2) memory, so PMC-GP uses a sparse variational approximation for the tested sample sizes.The common settings use 64 fixed inducing inputs, and PMC estimates and uncertainty come from the marginal q(f_h(0_m)); experiment-specific deviations are listed separately.Source: §C.2 · p. 58, §C.3 · p. 59

C.4. Details of Main-Text Experiments and Diagnostics

The appendix specifies protocols for diagnostic and benchmark experiments, then documents real-data and protein-image setups and hyperparameter selection.

C.4. Details of Main-Text Experiments and Diagnostics

  • The PPC compares a misspecified linear conditional-mean fit with a well-specified quadratic fit using 1,000 posterior-predictive draws and 10 discrepancy bins.The check refits the polynomial model and recomputes the test statistic for replicated response datasets.Source: §C.4 · p. 59
  • Dimension scaling fixes the estimand as dimension changes; the overlap study varies µ from 0 to 6 and reports median latent GP posterior variance on a log scale.The reported variance excludes observation noise.Source: §C.4 · pp. 60, 62, 64
  • Calibration constructs Gaussian intervals at nominal levels from 0.05 to 0.95 and assesses their empirical coverage over replicates.The passages describe the evaluation procedure, not its resulting coverage values.Source: §C.4 · p. 63

C.5. Fusion Methods and Oracle

  • Fusion benchmarks use 100,000 output particles, 20 paired replications, shared component samples, and Poisson(S) draws per component.The supplementary suite uses S = 10^5 in the univariate experiment as well.Source: §C.5 · p. 65

C.6. Real-Data Experiments

  • Real-data logistic regressions cover respiratory, HIGGS, and MAGIC; GBF was not run for HIGGS because it was computationally infeasible at this scale.The respiratory data comprise two clinical-center batches, while HIGGS and MAGIC each use ten batches.Source: §C.6 · p. 65

C.7. Simulation-Based Inference

  • For HSP90, one amortized NPE-C network is conditioned separately on ten standardized images, and PMC estimates the first two moments against a product-posterior reference.The reference is the normalized product of the ten fitted image-level subposterior densities.Source: §C.7 · p. 66

C.8. Selection of the Hyperparameters

  • Experiment-level settings were selected manually, with specified large simulation cells receiving individual overrides assessed in paired comparisons.The overrides cover large sample sizes with 20 components, variance scaling or imbalance, and the largest location-and-scale-grid cell.Source: §C.8 · p. 67
  • PMC-MLP training steps grow with the square root of realized sample size, from 200 steps at S ≤ 100 to an 8,000-step cap at S = 160,000.The realized sample size is drawn from a Poisson distribution with mean equal to the nominal sample size.Source: §C.8 · pp. 67–68
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