Amortized Data Borrowing with Exchangeability-Aware Neural Posterior Estimation

Chin-Hung Huang; JooChul Lee; Huan He

cs.LGSep 30, 2026

Amortized Data Borrowing with Exchangeability-Aware Neural Posterior Estimation

Authors: Chin-Hung Huang, JooChul Lee, Huan He

Organizations: Department of Mathematics and Statistics Auburn University

Abstract

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-.

Figures & tables

Figure 1: Proposed NPE workflow. Simulation scenarios generate current/external pairs for pretraining; at inference, the same summaries from a new pair are passed through the pretrained NPE to return a posterior for the current-study target.

Scenario	Covariates	Outcomes
SC1	exchangeable	exchangeable
SC2	partial shift	exchangeable
SC3	exchangeable	partial drift
SC4	strong shift	exchangeable
SC5	exchangeable	strong drift
SC6	strong shift	strong drift

Table 1: Two-axis taxonomy of source-mismatch regimes. SC1 is the exchangeable baseline; SC2 and SC4 isolate covariate shift at increasing strength; SC3 and SC5 isolate outcome drift at increasing strength; and SC6 combines strong covariate shift with strong outcome drift.

Scenario	Method	Bias	Bias SE	Cov.	Type I	Type I SE
SC3	PSPower	0.96	0.02	0.00	1.00	0.00
SC3	IW	0.66	0.02	0.00	1.00	0.00
SC3	Commensurate	0.86	0.02	0.00	1.00	0.00
SC3	NPE	0.15	0.02	1.00	0.02	0.02
SC5	PSPower	0.95	0.01	0.00	1.00	0.00
SC5	IW	0.66	0.02	0.00	1.00	0.00

Table 2: Representative hard-setting summary (

N_{c}=100

). Bias and coverage are from

\theta=1

; Type I error is from

\theta=0

Figure 2: Non-null estimation error by scenario. Each spoke is one simulation scenario; values closer to the center indicate smaller error.

Figure 3: Time to a high-quality joint posterior. Per-dataset wall-clock (log scale) required to reach

\min(\mathrm{ESS}(\theta),\mathrm{ESS}(\phi))\geq 1000

for NPP and Commensurate; NPE is timed on its single amortized forward pass. Boxes span the IQR across

B=5

replications

\times

four distribution-shift scenarios. NPE is

\sim\!10^{3}\times

faster than either MCMC sampler and uniformly so across regimes.

Cohort	Number of subjects	Conversion rate
Concurrent (ADNIGO + ADNI2)	376	0.1702
External (ADNI1)	337	0.3769

Table 3: ADNI cohort summary for the real-data borrowing example.

Method	AUC	Accuracy	Brier	LogLoss	MeanPred	MeanObs
ConcurrentOnly	0.918	0.881	0.084	0.271	0.183	0.168
ExternalOnly	0.923	0.887	0.088	0.303	0.248	0.168
Pooled	0.923	0.883	0.083	0.276	0.218	0.168

Table 4: Baseline benchmark on a held-out concurrent test set.

Quantity	Value
Posterior mean of $\theta$	$-1.5990$
Posterior SD of $\theta$	$0.8585$
95% interval for $\theta$	$(-3.3009,\ 0.1082)$
Observed concurrent event rate	$0.1702$
Observed external event rate	$0.3769$
Empirical concurrent logit	$-1.5841$

Table 5: Main ADNI real-data result for the NPE-based formulation.

Appendix figures & tables8 assets

Supplementary material from the paper’s appendix.

Appendix

$N_{c}$	Scenario	Method	Bias	RMSE	Coverage	Power
50	SC1	PSPower	0.0183	0.1268	0.90	1.00
	SC1	IW	0.0365	0.1309	0.94	1.00
	SC1	Commensurate	0.0183	0.1234	0.90	1.00
	SC1	NPE	0.2201	0.3049	0.94	1.00
	SC2	PSPower	-0.0126	0.1485	0.94	1.00
	SC2	IW	-0.0223	0.1505	0.98	1.00

Appendix

Table 6: Full non-null simulation results (

\theta=1

Scenario	Method	Bias	RMSE	Coverage	Type I
$N_{c}=50$
SC1	PSPower	0.0187	0.1269	0.90	0.10
SC1	IW	0.0373	0.1311	0.94	0.06
SC1	Commensurate	0.0187	0.1235	0.90	0.10
SC1	NPE	0.3172	0.3811	0.90	0.10
SC2	PSPower	-0.0117	0.1485	0.94	0.06

Appendix

Table 7: Full null simulation results (

\theta=0

Figure 4: Additional non-null operating characteristics. Each spoke is one simulation scenario; the dashed polygon marks nominal 95% coverage when applicable.

Figure 5: Null operating characteristics by scenario. Dashed polygons indicate the nominal benchmark for each metric: 0.95 for coverage and 0.05 for Type I error.

Noise SD	Bias	Absolute bias	RMSE	Coverage
0.00	0.222	0.222	0.273	0.98
0.02	0.226	0.226	0.276	0.98
0.05	0.241	0.241	0.290	0.98
0.10	0.270	0.270	0.328	0.98
0.15	0.296	0.296	0.357	0.96

Appendix

Table 8: Sensitivity to perturbing the OLS-derived coefficient-difference summary block in SC5 (

n_{c}=100

\theta=1

Scenario	Method	Bias	RMSE	Coverage	Width
SC1	Concurrent OLS	-0.006	0.134	0.94	0.538
SC1	NPE- $\beta_{1}$	0.002	0.082	1.00	0.442
SC2	Concurrent OLS	0.003	0.136	0.92	0.542
SC2	NPE- $\beta_{1}$	0.016	0.064	1.00	0.459
SC3	Concurrent OLS	-0.002	0.142	0.94	0.530
SC3	NPE- $\beta_{1}$	0.004	0.069	0.98	0.434

Appendix

Table 9: Limited illustrative check for a current-study covariate-effect target,

\beta_{1}

(

n_{c}=100

n_{e}=200

). NPE uses target-aware summaries and validation-based posterior-scale calibration.

$\mathrm{sd}(\delta)$	Posterior mean	Mean SD	95% interval	Width
0.50	-1.573	0.864	$(-3.281,\ 0.122)$	3.403
0.75	-1.583	0.862	$(-3.287,\ 0.107)$	3.394
1.00	-1.587	0.856	$(-3.279,\ 0.092)$	3.371
1.25	-1.592	0.854	$(-3.279,\ 0.082)$	3.360

Appendix

Table 10: ADNI sensitivity to the prior standard deviation for the external–concurrent outcome-shift parameter

\delta

Scenario	Method	Bias	RMSE	Power
SC3	NPE	0.1320	0.2223	1.00
SC3	KL-NPE	0.0242	0.1905	0.98
SC3	OT-NPE	0.0757	0.2013	1.00
SC3	KL+KS-NPE	0.0964	0.2037	1.00
SC3	JSD-NPE	0.1769	0.2498	1.00
SC5	NPE	0.1483	0.2205	1.00

Appendix

Table 11: Exploratory NPE extensions in the harder non-null settings (

N_{c}=100

\theta=1

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