NP-LEAP: Nonparametric Latent Exchangeability Prior for Model-Lean Borrowing from Historical Data

Abstract: Bayesian dynamic borrowing (BDB) methods leverage historical data to reduce treatment effect uncertainty, yet existing approaches rely on parametric outcome models susceptible to misspecification. We propose the nonparametric latent exchangeability prior (NP-LEAP), an outcome-agnostic, assumption-lean framework to borrow information from historical data. The NP-LEAP performs individual-level exchangeability assessment, inducing Bayesian model averaging over all possible partitions of the historical data into exchangeable and nonexchangeable subsets. Although applicable to a variety of data types with choice of appropriate kernel, the NP-LEAP is particularly well-suited for studies with time-to-event outcomes, where parametric BDB is potentially triply misspecified - imposing a parametric baseline hazard, the proportional hazards structure, and blanket exchangeability. We establish posterior consistency under mild regularity conditions. Simulation studies demonstrate favorable operating characteristics relative to parametric borrowing methods and nonborrowing semiparametric frequentist methods. We illustrate the method by augmenting the control arm in a randomized trial of patients with non-small cell lung cancer.

• The paper presents NP-LEAP, a novel Bayesian modeling framework that improves on LEAP by using Dirichlet process mixture models, allowing for model lean dynamic borrowing from historical data without parametric assumptions.
• NP-LEAP offers interpretability and transparency through an exact Bayesian model averaging (BMA) framework, making it easier to understand the contribution of historical data.
• The methodology includes a fully tractable Gibbs sampler and has demonstrated robustness in practical implementations.

Motivation and overview

NP-LEAP extends the Latent Exchangeability Prior (LEAP) framework (2608.16688) for Bayesian dynamic borrowing from historical data by replacing parametric density assumptions on both the current-data distribution and the non-exchangeable component of the historical-data distribution with Dirichlet process mixture models. The motivating problem is standard in regulatory settings: a current trial with $n$ subjects is supplemented by historical control data of size $n_0$, where an unknown fraction $\gamma$ of historical individuals is exchangeable with the current population. The LEAP assigns each historical subject a latent Bernoulli exchangeability indicator $\epsilon_{0j} \sim \text{Ber}(\gamma)$; NP-LEAP's contribution is to make the exchangeable density $f$ and the non-exchangeable density $g$ arbitrary objects governed by Bayesian nonparametric priors, so that borrowing is "model-lean" — it does not depend on a correctly specified parametric family.

BMA interpretation

The paper establishes that the nonparametric extension of the LEAP admits an exact Bayesian model averaging (BMA) interpretation. Under independent priors $\Pi(df)\Pi(dg)\Pi(d\gamma)$, the joint posterior factorizes as a product of the conditional posteriors of $f$ given the exchangeable subsets $(\bm{y}, \bm{y}_{0,\text{exch}})$, of $g$ given the non-exchangeable subset, and of the partition probabilities:

$n_0$0

This result has a practical implication: the marginal prior probability of any exchangeability pattern, $n_0$1, acts as the model weight in the average, so the induced shrinkage of historical information is transparent and interpretable.

Clustering scheme and MCMC implementation

To make the model computable, the authors take finite-mixture limits ($n_0$2) of two-part mixtures of Dirichlet mixture models, deriving a Gibbs sampler whose full conditionals are available in closed form. Three structural results underpin the algorithm. First, the prior clustering mechanism for current-data individuals reduces exactly to the standard "rich get richer" Chinese restaurant process with concentration $n_0$3. Second, historical individuals are assigned jointly to a class ($n_0$4) and a cluster: an exchangeable assignment joins existing current/historical clusters with probability proportional to cluster occupancy, while a non-exchangeable assignment enters an independent DP($n_0$5) clustering over the non-exchangeable component only. Third, all full conditionals are tractable: $n_0$6 follows a spike-and-slab form mixing a point mass at $n_0$7 (with weight $n_0$8) with a Beta$n_0$9 distribution when no non-exchangeable individuals remain; the DP concentrations $\gamma$0 and $\gamma$1 have Gamma conditionals; stick-breaking variables have independent Beta conditionals; and cluster parameters update from within-cluster posteriors pooling current and exchangeable-historical observations.

The point mass at $\gamma$2 is notable because it permits complete borrowing when the historical data show no evidence of conflict — a feature the authors argue avoids artificial discounting under full exchangeability. The base measure for the non-exchangeable component is deliberately diffuse (zero mean, large variance), which the paper states prevents spurious preference toward assigning clusters to the exchangeable group.

For survival applications, the kernel is instantiated as an ANOVA-dependent Dirichlet process (DDP) log-normal AFT model, with base measures elicited from maximum likelihood fits discounted to an effective sample size of 2, and Gamma base measures for precisions chosen by KL-divergence minimization to preserve semi-conjugacy and computational speed.

Posterior consistency

The theoretical core of the paper is an asymptotic consistency theorem. Under a product prior satisfying KL-support and exponentially decaying sieve conditions — verified explicitly for Dirichlet process location-scale mixtures of Gaussians with scalar-identity covariance kernels and mild base-measure regularity — the LEAP posterior concentrates around the true parameter $\gamma$3 in a weighted Hellinger pseudo-metric $\gamma$4 that weights the current density by the limiting current-data fraction $\gamma$5:

$\gamma$6

A corollary shows Hellinger consistency at $\gamma$7, the density of primary inferential interest, whenever $\gamma$8. The proof combines the standard Schwartz-type argument (Jensen's inequality for the denominator lower bound plus testing-based upper bounds over entropy balls and the sieve complement) with a product-space entropy bound showing that covering the product space costs only the sum of the component entropies plus a logarithmic term for discretizing $\gamma$9.

An important concession accompanies this theorem: the LEAP model is inherently non-identifiable. Any pair $\epsilon_{0j} \sim \text{Ber}(\gamma)$0 satisfying $\epsilon_{0j} \sim \text{Ber}(\gamma)$1 yields identical likelihoods, so consistency can only hold up to the pseudo-metric equivalence class, not at $\epsilon_{0j} \sim \text{Ber}(\gamma)$2 itself. The authors note that the exceptional set has zero prior mass but leave open whether stronger identifiability conditions could yield consistency at the full parameter vector. Consistency is also stated without rates; convergence-rate results under the mixture structure remain unestablished.

Simulation design and tipping point analysis

The simulation study mimics a real oncology hybrid-control setting with time-to-event outcomes generated from a spline-based proportional hazards model fit to pooled real data, with treatment effects $\epsilon_{0j} \sim \text{Ber}(\gamma)$3 and $\epsilon_{0j} \sim \text{Ber}(\gamma)$4 on the log-cumulative-hazard scale. Current data use $\epsilon_{0j} \sim \text{Ber}(\gamma)$5 with permuted-block randomization; dropout is exponential with arm-specific rates calibrated so approximately 5% drop out within one year. Historical data are generated under three exchangeability regimes ($\epsilon_{0j} \sim \text{Ber}(\gamma)$6) crossed with three degrees of drift ($\epsilon_{0j} \sim \text{Ber}(\gamma)$7 applied to the historical effect). To mimic regulatory constraints, the design fixes a single historical dataset per scenario, selected among 100,000 replicates by closest agreement between its Kaplan–Meier estimator and the data-generating process — a choice that makes results conditional on one realized history rather than averaged over histories.

For regulatory practice, the paper demonstrates a tipping point analysis enabled directly by the latent-$\epsilon_{0j} \sim \text{Ber}(\gamma)$8 construction: fixing $\epsilon_{0j} \sim \text{Ber}(\gamma)$9 over a grid from 0 to 1 and examining the posterior for the difference in 12-month survival probabilities. In the application shown, the 95% credible interval excludes zero only at $f$0, meaning conclusions survive nearly complete borrowing. The authors further observe an asymmetric sensitivity — the interval's lower bound is robust while the upper tail shifts sharply for $f$1 — which they attribute to separation between the historical controls below 12 months and congruence above it; changing the estimand to 24 months alters the lower bound as well.

Limitations and open questions

Several constraints qualify the results. The consistency theory covers abstract densities and does not address the censoring mechanism or the DDP regression structure actually used in applications; extending the argument to censored, covariate-dependent settings remains open. The non-identifiability of $f$2 means posterior summaries of $f$3 itself require careful interpretation, and the paper provides no convergence-rate guarantees. Computationally, the Gibbs sampler's categorical reassignment steps scale linearly in the number of observations per iteration, but mixing behavior under near-complete separation of historical data is not formally studied. Finally, the tipping point illustration rests on a single fixed historical dataset, so operating characteristics across repeated histories — particularly calibration of the spike weight $f$4 — are not characterized.

Conclusion

NP-LEAP supplies a nonparametric, computationally explicit generalization of the latent exchangeability prior, with a Gibbs sampler admitting closed-form updates including a spike-and-slab prior on the exchangeability probability that permits exact borrowing, a proof of posterior consistency under verifiable conditions on DP Gaussian mixture priors, and a demonstration that the latent-indicator structure supports regulator-facing tipping point analyses. Its main unresolved issues are identifiability beyond the pseudo-metric equivalence class, consistency under censoring and regression structure, and rate optimality.