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Amortized Data Borrowing with Exchangeability-Aware Neural Posterior Estimation

Chin-Hung Huang, JooChul Lee, Huan He

arXiv:2609.38902v1cs.LGstat.ML

TL;DR

Small concurrent studies may be underpowered despite related external data, while classical BDB relies on low-dimensional compatibility summaries and repeated posterior fitting. This paper develops summary-based amortized NPE across cohort pairs, with its strongest reported robustness in harder mismatch settings, especially outcome drift.

  • Problem

    Small concurrent studies can be underpowered despite available related cohorts, while classical BDB compresses compatibility into low-dimensional quantities and repeatedly fits posteriors.

    Source: §1 · p. 1, §2.1 · p. 6

  • Method

    The method amortizes summary-based NPE across simulated current/external cohort pairs, using mismatch-organized summaries so one pretrained network can be reused.

    Source: §1 · pp. 3–4, §3.3 · p. 9

  • Results

    NPE is most robust in harder mismatch settings, especially outcome-mechanism drift, and in the ADNI example recovered concurrent-cohort risk more closely than naive borrowing strategies.

    Source: §1 · p. 4

  • Takeaways & Limitations

    Mismatch-specific summary blocks provide an audit layer for checking cohort-pair signals before trusting a posterior.

    Source: §3.3 · p. 9

    Posterior quality depends on the pretraining simulator, so mismatch patterns outside its support cannot be handled reliably.

    Source: §6 · p. 16

Abstract

from arXiv · show

Augmenting small concurrent studies with external or historical cohorts is attractive in drug development, where enrollment is slow, follow-up is expensive, and closely related trial or real-world data are often already available. Bayesian dynamic borrowing (BDB) provides a principled framework for adaptively controlling the influence of external data, but classical implementations often depend on hand-specified priors and MCMC-based inference, which can be computationally expensive and not generalizable. In this work, we study amortized neural posterior estimation (NPE) as a flexible alternative. A single network is pretrained on simulated current/external dataset pairs spanning covariate shift, outcome drift, and joint non-exchangeability, and then returns an approximate posterior for a scalar current-study target in a single forward pass. Through simulation studies, we find that NPE is most useful under outcome drift and joint mismatch: in the harder outcome-drift regimes, it gives up to about five-fold lower absolute bias than the best classical baseline and keeps Type I error close to nominal. After pretraining, posterior summaries are obtained in about 8 ms per dataset, roughly $10^3\times$ faster than MCMC-based borrowing baselines in our timing experiment. We further analyze Alzheimer's Disease Neuroimaging Initiative (ADNI) data and show that, when mild cognitive impairment outcomes differ across cohorts, the NPE formulation recovers the later-cohort risk level in this example without claiming greater precision. Code is available at https://github.com/ChinHungScott/NPE-for-Bayesian-Dynamic-Borrowing-MLHC-.

1. Introduction

This paper presents summary-based neural posterior estimation for borrowing across current and external cohorts, addressing the limits of scalar compatibility measures and repeated posterior fitting. It evaluates the method across simulated exchangeability regimes and in an ADNI cohort-shift example.

1. Introduction

  • Small or slow-accruing studies often lack statistical power even when related historical or external data exist, but borrowing safely depends on avoiding an assumption of full exchangeability.When exchangeability fails, naive pooling can induce bias and inflate downstream type I error.Source: §1 · pp. 1–2
  • Classical borrowing methods use low-dimensional compatibility measures that can conflate covariate shift with outcome drift, require prior or hyperparameter choices, and often require a fresh MCMC fit for each pair.Reweighting methods address covariate shift but do not correct shifts in the outcome mechanism itself.Source: §1 · pp. 2–3
  • The method pretrains one network on simulated current/external pairs and reuses it across pairs, replacing repeated per-dataset posterior fitting with amortized inference.The target is scalar: an additive study effect in Gaussian simulations and a cohort-level risk parameter in ADNI.Source: §1 · pp. 3–4
  • Across six exchangeability regimes, NPE’s main strength is robustness in harder mismatch settings, especially outcome drift, where compatibility summaries and covariate-balancing weights tend to struggle.The evaluation benchmarks representative Bayesian dynamic borrowing priors and propensity-based reweighting baselines.Source: §1 · p. 4
  • In ADNI, NPE can recover concurrent-cohort risk more closely than naive borrowing; external-only models rank subjects well but systematically overestimate concurrent-cohort risk.The case study uses baseline mild cognitive impairment subjects and 24-month conversion to dementia as the outcome.Source: §1 · pp. 3–4, Generalizable Insights about Machine… · p. 4

Generalizable Insights about Machine Learning in the Context of Healthcare

  • A combined simulation and case-study design assesses borrowing methods under known exchangeability violations and checks their behavior on clinically meaningful data.The paper frames real-world borrowing mismatch as potentially involving calibration rather than discrimination.Source: Generalizable Insights about Machine… · p. 4

2. Related Work

Prior work adapts external-data borrowing to exchangeability, while weighting methods address covariate differences and neural posterior estimation offers amortized inference with richer summaries. The paper situates its approach against the distinct limitations of these methods.

2.1. Bayesian dynamic borrowing and the exchangeability assumption

  • Dynamic borrowing adjusts the contribution of external data according to cross-source compatibility, but classical methods may compress distinct shifts into a few quantities and require repeated posterior fitting.Exchangeability supports precision gains through pooling, whereas its failure can induce bias and inflate type I error.Source: §2.1 · pp. 5–6

2.2. Balancing and reweighting methods

  • Weighting and balancing methods adjust external cohorts toward the current cohort’s covariate distribution, but they do not address differences in the outcome-generation mechanism.The paper identifies calibration-level mismatch, rather than covariate imbalance, as the dominant failure mode in its ADNI case study.Source: §2.2 · p. 6

2.3. Neural posterior estimation and simulation-based inference

  • Neural posterior estimation learns a conditional posterior from simulated parameter–data pairs and can reuse inference across new pairs with summaries richer than one compatibility scalar.For borrowing, the representation must preserve both target-estimand information and structure indicating when external data should influence inference.Source: §2.3 · pp. 6–7

3. Method

The method targets a scalar parameter in the concurrent population and uses a pretrained neural density estimator to approximate its posterior from paired current and external data. Its summary features distinguish types of source mismatch, while simulated pretraining determines the mismatch patterns the estimator can handle.

3.1. Setup and inferential target

  • Classical borrowing targets a scalar concurrent-population parameter θ and represents external-data influence through a borrowing-strength parameter ϕ.Examples include the power-prior discount a0 and commensurability precision τ.Source: §3.1 · p. 7

3.2. Amortized posterior inference

  • A single conditional density estimator maps each plausible dataset pair to an approximate posterior for θ, requiring only a summary calculation and one forward pass at inference.The method uses a Gaussian head as a deliberate simplification; richer density families yielded no measurable gains in preliminary experiments.Source: §3.2 · p. 8

3.3. Summary statistics

  • The exchangeability-aware summary combines source-specific moments, sample sizes, and separate OLS coefficients and residual variances to capture marginal shifts and outcome-mechanism drift.The coefficient gap Δβ is intended to signal outcome-mechanism drift not visible in marginal summaries.Source: §3.3 · pp. 8–9
  • Propensity-score overlap summaries and covariate mean gaps and Wasserstein-1 distance add signals for covariate overlap and joint mismatch.These diagnostics are designed to complement the marginal and conditional blocks; the blocks are expected to read small under exchangeability.Source: §3.3 · p. 9

3.4. Pretraining via simulated borrowing scenarios

  • Pretraining uses simulated current/external pairs across plausible mismatch conditions, and the resulting estimator is informed only by patterns within the simulator’s support.The simulator is not fit to the analysis data; its ranges encode assumptions about the target, nuisance parameters, covariate shift, and outcome drift.Source: §3.4 · pp. 9–10

4. Simulation Study

Across the simulated mismatch regimes, NPE is most useful under outcome drift or joint shift, while classical methods often perform well in easier settings; after pretraining, NPE produces posterior summaries much faster than MCMC-based borrowing.

4.1. Design

  • The Gaussian simulations vary covariate shift and outcome drift separately or together, train across their continuous ranges, and evaluate six anchor scenarios with Nc ∈ {50, 100} and Ne = 200.Source: §4.1 · p. 10

4.2. Methods compared

  • The comparison includes PSPower, individualized weighting, Commensurate, and NPE; the study expects NPE to help most under outcome drift or joint covariate/outcome shift.Source: §4.2 · p. 11

4.3. Evaluation metrics

  • For θ = 1, the study reports bias, RMSE, coverage, and power; for θ = 0, it reports coverage and Type I error, emphasizing the latter measures in the main text.Source: §4.3 · p. 11

4.4. Main results for the non-null setting

  • In SC3 and SC5, classical borrowing has large bias and essentially no coverage while NPE stays closer to the target; in SC6, NPE is more stable than PSPower and Commensurate and close to IW.Figure 2 shows the pattern across scenarios; for Nc = 100, Table 2 summarizes harder settings and its MCSEs indicate large method differences are not simulation noise.Source: §4.4 · pp. 11–12, Table 2 · p. 11, Figure 2 · p. 12

4.5. Null results

  • In the null run, classical methods become anti-conservative in SC3, SC5, and SC6, whereas NPE stays much closer to nominal Type I error.In SC1 and SC2, classical methods have cleaner operating characteristics.Source: §4.5 · p. 12

4.6. Computational timing

  • NPE returns marginal-θ posteriors in about 8 ms per dataset, versus 1.6–3.3 s median for Commensurate to meet the ESS target; NPP exhausts 8.6 s without reaching it.The timing comparison evaluates wall-clock time to a high-quality joint posterior against a single amortized NPE forward pass.Source: §4.6 · p. 12, Figure 3 · p. 13

4.7. Interpretation

  • NPE is not uniformly best: classical methods are often more efficient in easier regimes, while its advantage appears under outcome drift or joint shift, where fixed borrowing rules can yield large bias, poor coverage, and inflated Type I error.Source: §4.7 · p. 13

5. Real-Data Example: ADNI

The ADNI example treats borrowing as a cohort-risk problem, not a treatment-effect analysis. NPE estimates the concurrent cohort’s risk level, while cohort comparisons show why direct borrowing may be problematic.

5. Real-Data Example: ADNI

  • ADNI is used to examine whether earlier-cohort borrowing shifts inference from the later concurrent risk level, not to analyze a randomized treatment effect.The study presents the observed-data example as distinct from the simulations’ controlled methodological evaluation.Source: §5 · pp. 13–14

5.1. ADNI setup

  • The analysis compares baseline-MCI subjects in external ADNI1 with pooled ADNIGO and ADNI2 subjects, using conversion to dementia or Alzheimer’s disease by about 24 months.Table 3 summarizes the cohorts; the external cohort has a higher conversion rate than the concurrent cohort.Source: §5.1 · p. 14, Table 3 · p. 14

5.2. Baseline benchmark

  • ExternalOnly and Pooled predicted risks too high for the concurrent cohort, despite similar AUC and accuracy across all three baselines.The benchmark evaluated ConcurrentOnly, ExternalOnly, and Pooled logistic-regression models on a held-out concurrent test set.Source: Table 4 · p. 14, §5.2 · p. 14

5.3. ADNI-specific NPE formulation

  • NPE pretraining allows an external–concurrent outcome shift while targeting the concurrent risk level, rather than forcing full exchangeability.The shift parameter is used during pretraining but is not reported as a target estimand.Source: §5.3 · p. 15

5.4. ADNI results

  • The NPE posterior mean was −1.5990 versus empirical concurrent logit risk −1.5841, consistent with recovering later-cohort risk here but not greater precision.The posterior standard deviation was 0.8585 and the 95% interval was (−3.3009, 0.1082).Source: Table 5 · p. 15, §5.4 · p. 15

6. Discussion

The discussion presents NPE as amortized inference for scalar current-study targets and identifies simulator support and summary design as constraints on its scope.

6. Discussion

  • The empirical scope is a scalar current-study target: an additive study-effect parameter in Gaussian simulations and a cohort-level risk parameter in ADNI.Covariate effects, treatment–covariate interactions, and other causal functionals require additional evaluation; the covariate-effect check is only illustrative.Source: §6 · p. 16
  • The approach pretrains once on a broad simulation distribution and then obtains posteriors for new current–external pairs in a single forward pass.The discussion frames the ADNI example around its analyzed estimand rather than as a randomized treatment-effect study.Source: §6 · p. 16
  • Posterior quality depends on the pretraining simulator, so mismatch patterns outside its support cannot be handled reliably.The hand-crafted summaries may also miss richer patient-level structure.Source: §6 · p. 16

A.1. Full Simulation Tables

The appendix identifies full tables for non-null and null simulation results, without passage text reporting their numerical findings.

A.1. Full Simulation Tables

  • Table 6 contains full non-null simulation results for θ = 1, with column headings for bias, RMSE, coverage, and power.The supplied passages do not provide table cell values.Source: Table 6 · p. 21, §A.1 · p. 22
  • Table 7 contains full null simulation results for θ = 0, with column headings for bias, RMSE, coverage, and Type I error.The supplied passages do not provide table cell values.Source: Table 7 · p. 22, §A.1 · p. 23

A.2. Additional Simulation Figures

The appendix figures display additional non-null and null operating characteristics by simulation scenario, with dashed polygons marking nominal benchmarks.

A.2. Additional Simulation Figures

  • Figure 4 shows additional non-null operating characteristics, with each spoke representing a simulation scenario and a dashed polygon marking nominal 95% coverage when applicable.Full numerical results are identified as Tables 6 and 7.Source: Figure 4 · p. 23, §A.2 · p. 23
  • Figure 5 shows null operating characteristics by scenario, with dashed polygons marking 0.95 coverage and 0.05 Type I error benchmarks.Source: Figure 5 · p. 24

A.3. Implementation and Training Details

The NPE model uses a deterministic forward pass to produce a scalar-target posterior from precomputed summaries. Its pretraining samples mismatch types and cohort sizes across specified distributions.

A.3. Implementation and Training Details

  • At inference, a fully connected network with [128, 128, 64] hidden units and a Gaussian output head returns the posterior mean and log-standard-deviation in one deterministic forward pass.Dropout is used during training and disabled at inference; richer scalar-output density heads did not improve calibration or operating characteristics in these experiments.Source: §A.3 · p. 24
  • Pretraining samples θ, β, σ, concurrent covariate location, shift magnitudes, and mismatch type; external sample size is fixed at 200 and concurrent size varies across 50, 100, 200, and 500.The main evaluation grid uses concurrent sample sizes 50 and 100 in regimes SC1–SC6.Source: §A.3 · p. 24
  • The synthetic model receives estimated coefficient differences from separate concurrent and external OLS fits, while the true β is used only for data generation and operating-characteristic evaluation.Source: §A.3 · pp. 24–25

A.4. Perturbing the OLS-Derived Summary Block

In the SC5 outcome-drift setting, perturbing the OLS-derived coefficient-difference summaries left coverage at or above nominal through noise standard deviation 0.15, while bias rose gradually.

A.4. Perturbing the OLS-Derived Summary Block

  • Coverage stayed at or above nominal 95% for noise standard deviations up to 0.15, while absolute bias increased from 0.22 to 0.30.This sensitivity experiment perturbed the OLS-derived coefficient-difference block in SC5 with nc = 100; Table 8 reports 50 simulation replicates.Source: §A.4 · p. 25, Table 8 · p. 25
  • The experiment added independent Gaussian noise to the coefficient-difference block before standardizing the summary vector and applying the pretrained network.Source: §A.4 · p. 25
  • The coefficient-difference block is informative, but it is not the sole source of the network’s robustness under outcome drift.Source: §A.4 · p. 25

A.5. Illustrative Covariate-Effect Target Check

These checks examine target-aware summaries for a covariate effect, sensitivity to the ADNI outcome-shift prior scale, and exploratory discrepancy-feature variants in harder mismatch settings.

A.5. Illustrative Covariate-Effect Target Check

  • The illustrative β1 check augments summaries with source-specific OLS slopes and standard errors and calibrates posterior scale; it supports adapting the framework to a different scalar target, not full validation of arbitrary effects or interactions.A systematic study of covariate-effect targets and overlap requirements remains future work.Source: §A.5 · p. 25, Table 9 · p. 26

A.6. ADNI Prior-Scale Sensitivity

  • Across prior scales for the ADNI external–concurrent outcome-shift parameter δ, the posterior mean stayed close to the empirical concurrent logit risk of −1.584 and interval width changed only modestly.Each Table 10 row averages five independent NPE pretraining runs using the same observed covariates and outcome definition.Source: §A.6 · p. 26, Table 10 · p. 26

A.8. Exploratory NPE Extensions

  • Exploratory variants added KL, OT, KS, KL+KS, or JSD discrepancy summaries and were assessed in SC3 and SC5 outcome-drift settings and SC6 joint shift; Table 11 gives representative results for Nc = 100.These variants were intended to test whether discrepancy features reduce bias or capture a missing mismatch signal; they are not part of the main comparison.Source: §A.8 · p. 27, Table 11 · p. 27
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