PPD-CPP: Dynamic Borrowing in Bayesian Analysis

• The paper introduces a dynamic calibration method that leverages predictive density congruence to adjust the power prior for optimal historical data borrowing.
• It quantifies heterogeneity through the congruence measure (p_CM) and maps deviations via a sigmoid function to determine the pointwise borrowing weight.
• Simulation studies and real data applications in clinical trials and toxicity tests demonstrate improved inferential precision and bias reduction.

The Pointwise Predictive Density Calibrated-Power Prior (PPD-CPP) is a Bayesian methodology for dynamically incorporating historical information into current analyses. It extends and generalizes the classical power prior by calibrating the degree of information borrowing at the individual data point level according to a congruence measure derived from posterior predictive densities, ensuring borrowing consistency and adaptive robustness in the presence of data heterogeneity or prior-data conflict. This framework has been developed with the explicit aim to address the challenges posed by dynamic borrowing in rare disease, clinical trials, and other contexts that require principled integration of external controls, as described in recent methodological work (Wang et al., 30 Sep 2025).

1. Conceptual Foundation of PPD-CPP

PPD-CPP builds on the standard power prior approach, in which historical data likelihood is raised to a power parameter $\alpha$, moderating its influence on the joint posterior: $$\pi(\theta \mid D^{c}, D^{h}, \alpha) \propto L(\theta \mid D^{c}) \; L(\theta \mid D^{h})^{\alpha} \; \pi_{0}(\theta).$$ The innovation in PPD-CPP lies in replacing the fixed scalar $\alpha$ with a dynamically calibrated parameter (either global or covariate-specific), which is determined by the similarity—or congruence—between historical data $D^{h}$ and current data $D^{c}$. The congruence is quantified via a new estimand $p_{CM}$: the marginal posterior predictive $p$-value that evaluates how well historical data can replicate or explain the current observations.

Main objectives of PPD-CPP:

• Quantify heterogeneity between datasets using predictive distributions.
• Configure the level of historical data borrowing through a pointwise or individualized power parameter.
• Achieve borrowing consistency—full borrowing when data are congruent, minimal borrowing under pronounced discordance.

2. Congruence Measure ($p_{CM}$): Definition and Asymptotic Properties

A central methodological construct is the congruence measure $p_{CM}$, defined as follows: $$p_{CM} = P\left\{ T(y^{rep}) \geq T(y^{c}) \mid D^{h} \right\},$$ where $y^{rep}$ is a posterior predictive replicate generated from the historical data $D^{h}$, and $T(\cdot)$ denotes a test statistic. Two formulations are considered:

• $T(x) = x$, comparing observed values directly.
• $T(x; D^{h}) = p(x \mid D^{h})$, evaluating the marginal posterior predictive density at the current observation.

The likelihood-based version thus computes: $$p_{CM} = P\left\{ p(y^{rep} \mid D^{h}) \geq p(y^{c} \mid D^{h}) \right\}.$$ This probability provides an interpretable diagnostic for congruence: under model homogeneity, $p_{CM} \to 1/2$ as the historical sample size $m \to \infty$. In contrast, increasing the discrepancy in means or variances will asymptotically drive $p_{CM}$ toward $0$ (or $1$), establishing a theoretical foundation for calibrating the power parameter in data-adaptive fashion (Wang et al., 30 Sep 2025).

3. Dynamic Calibration of the Power Parameter

Calibrating the power prior parameter is achieved by mapping the deviation of $p_{CM}$ from $1/2$ onto $\alpha$ via a monotone transformation, typically a two-parameter sigmoid function: $$S = |p_{CM} - 1/2|,\qquad \alpha = g(S) = \frac{1}{1 + \exp(a + b \cdot \log S)},$$ with $a \in \mathbb{R}$ and $b > 0$. The function is pre-calibrated—outside of the current data—so that $\alpha$ is close to $1$ under high congruence ($S \to 0$), and approaches $0$ under strong incongruence ($S \to 1/2$).

Extension to regression frameworks is accomplished by computing $p_{CM,i}$ for each data point $i$, permitting individual or covariate-specific $\alpha_i$ assignments: $$\alpha_{i} = 1 / [1 + \exp\{a + b \cdot \log(|p_{CM,i} - 1/2|)\}].$$ Algorithmic implementation involves Bayesian posterior simulation for predictive densities, tabulation of $p_{CM}$ (or $p_{CM,i}$), and mapping to power parameters for posterior inference.

4. Borrowing Consistency and Theoretical Guarantees

Borrowing consistency asserts that dynamic calibration recovers the expected extremes in the limiting regimes. The formal result (Theorem 2) shows:

• If $D^{c}$ and $D^{h}$ are fully congruent, $p_{CM} \to 1/2$ and $\alpha \to 1$.
• If mean or variance discrepancies increase without bound, $p_{CM} \to 0$ (or $1$), and $\alpha \to 0$. This property ensures that the method borrows maximally when justified and discards historical information in cases of severe heterogeneity.

The framework enables both global calibration and individualized (“pointwise”) modeling, allowing for selective borrowing from subgroups or adjustment for covariate shifts in regression models.

5. Simulation Studies and Comparative Evaluation

Extensive simulation studies demonstrate the robustness and adaptivity of PPD-CPP under varied scenarios:

• With normally distributed historical and current data sharing parameters, $p_{CM}$ converges to $1/2$ and the power parameter approaches $1$, showing full borrowing.
• As the difference in means increases, $\alpha$ drops sharply, evidencing dynamic adjustment.
• The likelihood-based congruence measure yields more sensitive detection of incongruence than direct observation comparison.
• Comparisons to Kolmogorov–Smirnov-based calibration and elastic power prior (EPP) show improved bias and coverage probability in intermediate congruence zones.
• In the presence of covariate heterogeneity (regression), individualized $\alpha_i$ enables selective groupwise borrowing, mitigating the inferential risk from covariate shifts.

6. Applications to Real Data Analysis

The methodology is validated via two substantive case studies: (a) Mother’s Gift Study: In this randomized trial, information from one site (historical) is adaptively borrowed for analysis of another (current) site. Regression modeling with dynamic $\alpha$ produces more precise effect estimates without compromising inference robustness. Parameter precisions are increased (via reduced posterior variance) when estimated $\alpha$ is close to $1$.

(b) Ceriodaphnia dubia Toxicity Test: For dose-response analysis, historical control data are incorporated using PPD-CPP with Poisson regression. Monte Carlo approximation is employed for $p_{CM}$ due to the absence of closed-form expressions; the estimated $\alpha \approx 0.55$ leads to partial borrowing with substantial reduction in posterior uncertainty and interval lengths.

7. Technical and Implementational Considerations

PPD-CPP relies on high-fidelity Bayesian posterior simulation, including efficient computation of marginal posterior predictive densities, robust Monte Carlo or importance sampling for likelihood components, and careful calibration of sigmoid parameters. Pre-data calibration leverages theoretical and Monte Carlo analysis of Bernoulli indicator variables arising from the $p_{CM}$ derivation. Covariate-specific calibration is enabled by individual computation of predictive densities for each subject, supporting granular control over information transfer.

The method demonstrates strong theoretical guarantees, robust empirical performance, and flexibility for practical implementation in frequentist and Bayesian modeling platforms (Wang et al., 30 Sep 2025).

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PPD-CPP represents a principled advancement in dynamic information borrowing, combining predictive density diagnostics with flexible, pointwise calibration of the power parameter. Its congruence-measure-based adaptivity positions it as a rigorous tool for clinical trials, rare disease studies, and any context requiring judicious integration of historical or external data.