Non-Homogeneous Poisson Process (NHPP) Framework

• Non-Homogeneous Poisson Process (NHPP) framework is a probabilistic model for recurrent events that captures time-varying and covariate-dependent intensity.
• It employs Soft Bayesian Additive Regression Trees (SBART) to nonparametrically model smooth, nonlinear, and high-dimensional interactions between time and covariates.
• Bayesian computation via data augmentation and hierarchical priors ensures efficient inference with strong empirical validation in both simulations and real-world applications.

A non-homogeneous Poisson process (NHPP) framework provides a flexible, fully probabilistic paradigm for modeling and inference on recurrent event data where the conditional intensity function can vary with both time and covariates, capturing non-proportional, nonlinear, and potentially high-dimensional effects beyond the scope of classical survival analysis methods. Bayesian machine learning approaches, particularly Soft Bayesian Additive Regression Trees (SBART), have recently been developed to nonparametrically learn the NHPP intensity function in the presence of complex functional forms, subject-specific heterogeneity, and nonlinear covariate/time interactions (Chen et al., 10 Jun 2026).

1. NHPP Model Structure and Conditional Intensity Specification

The standard NHPP for recurrent events assumes that the counting process $N_i(t)$ for each subject $i\in\{1,\dots,n\}$ has conditional intensity

$$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$

where:

• $\lambda_0$ is a constant baseline hazard,
• $u_i$ is a latent subject-specific frailty ($u_i > 0$; typically Gamma-distributed for multiplicative random effects),
• $g(X_i, t)$ is an unknown function of covariates $X_i$ and time $t$ satisfying $g(X_i, t)>0$.

This formulation decouples baseline, subject-level, and time-varying/random effects, allowing $i\in\{1,\dots,n\}$0 to model arbitrary smooth or complex interactions between the covariates and time. A common SBART representation is

$i\in\{1,\dots,n\}$1

with $i\in\{1,\dots,n\}$2 a latent regression function modeled by a sum-of-trees prior, and $i\in\{1,\dots,n\}$3 the standard normal CDF so $i\in\{1,\dots,n\}$4 (Chen et al., 10 Jun 2026).

2. Nonparametric Modeling of NHPP Intensity via SBART

Instead of imposing a fixed functional form for $i\in\{1,\dots,n\}$5, SBART places an additive regression-tree prior on $i\in\{1,\dots,n\}$6. Each tree $i\in\{1,\dots,n\}$7 partitions the input $i\in\{1,\dots,n\}$8 space via soft, probabilistic splits: $i\in\{1,\dots,n\}$9 where $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$0 encodes the tree structure as a sequence of branch/leaf splits on joint time-covariate dimensions, and the soft path weights $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$1 are products of logistic (inverse-logit) gating functions. The full regression function is

$$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$2

and the transformed intensity is $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$3 (Chen et al., 10 Jun 2026).

Key attributes of the SBART construction in this context:

• Supports high-dimensional, nonlinear, and potentially nonstationary intensity surfaces.
• Each branch node’s softness parameter $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$4 controls the transition sharpness; as $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$5, the model recovers hard partitioning.
• The prior over trees follows a Galton–Watson process with depth-dependent split probabilities, split variable/uniform cut allocations, and Gamma priors for bandwidth.

3. Hierarchical Priors and Latent Structure

The NHPP-SBART framework utilizes several hierarchically structured priors:

• Frailty effects: $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$6 with hyper-prior $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$7, enforcing $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$8 for identifiability.
• Baseline hazard: $$\lambda_i(t \mid X_i, u_i) = \lambda_0 \cdot u_i \cdot g(X_i, t)$$9.
• Tree structure: per-node split with probability $\lambda_0$0 (standard choices: $\lambda_0$1, $\lambda_0$2) (Chen et al., 10 Jun 2026).
• Leaf parameters: $\lambda_0$3, typically with $\lambda_0$4 for $\lambda_0$5 trees.
• Bandwidths $\lambda_0$6: shared per tree as $\lambda_0$7 with $\lambda_0$8 user-chosen.

4. Bayesian Computation: Data Augmentation and Posterior Inference

Direct likelihood evaluation in the NHPP framework involves terms of the form $\lambda_0$9, which is analytically intractable with nonparametric $u_i$0. SBART employs a two-layer data augmentation:

Layer 1 (Poisson Thinning): For each individual, simulate pseudo-events $u_i$1 from an independent NHPP with intensity

$u_i$2

over $u_i$3. The likelihood, conditioned on observed events $u_i$4 and pseudo-events $u_i$5, factorizes into products over $u_i$6 and $u_i$7 (Chen et al., 10 Jun 2026).

Layer 2 (Latent Gaussian Variables): An Albert–Chib (probit) construction introduces latent $u_i$8: for each event $u_i$9, $u_i > 0$0 truncated above $u_i > 0$1 (event), for each pseudo-event $u_i > 0$2, $u_i > 0$3 truncated below $u_i > 0$4 (no event). This converts the backfitting update for $u_i > 0$5 into a Gaussian regression problem, directly compatible with standard SBART Metropolis–Hastings moves (Chen et al., 10 Jun 2026).

Conditional posterior updates for frailties, baseline, and tree parameters are all available in closed form or via standard Gibbs/slice/MH steps.

5. Theoretical and Empirical Properties

The SBART-based NHPP approach provides the following properties:

• Smoothness Adaptivity: SBART can adapt to arbitrary (unknown) levels of smoothness in $u_i > 0$6, with posterior contraction rates matching minimax optimality for Hölder classes (Linero et al., 2017).
• High-dimensional/Interaction Recovery: The additive sum-of-trees and Dirichlet splits induce automatic adaptation to sparsity, variable selection, and nonlinear interactions (Linero et al., 2017, Linero, 2022).
• Robustness under Misspecification: Simulation studies confirm that SBART-based NHPP estimation yields lower mean squared error for cumulative intensity estimation and better goodness-of-fit via Martingale residuals even when frailty or intensity model assumptions are violated. For instance, in scenarios with $u_i > 0$7 subjects and $u_i > 0$8 covariates, RecSBART yielded average MSEs of $u_i > 0$9–$g(X_i, t)$0 across a range of simulated settings, outperforming both random forest and proportional-frailty models (random forest, $g(X_i, t)$1–$g(X_i, t)$2; proportional-frailty, $g(X_i, t)$3–$g(X_i, t)$4) (Chen et al., 10 Jun 2026).
• Real-world validation: Applied to colorectal-cancer recurrent hospitalization data ($g(X_i, t)$5), RecSBART achieved lowest mean squared Martingale residual (MSMR: $g(X_i, t)$6) versus RecForest ($g(X_i, t)$7) and a standard Bayesian proportional frailty model ($g(X_i, t)$8). Subject-level cross-validation showed RecSBART generalization gap $g(X_i, t)$9 vs. $X_i$0 for RecForest (Chen et al., 10 Jun 2026).

6. Implementation, Scalability, and Practical Recommendations

The NHPP-SBART framework is implemented in compiled C++ code, enabling practical computation for moderate sample sizes (e.g., $X_i$1 takes $X_i$2s per MCMC iteration). No additional computational accelerations beyond the two-layer augmentation are required for fit or mixing. Default choices for SBART hyperparameters (e.g., $X_i$3, $X_i$4, $X_i$5, $X_i$6) are empirically robust. Posterior diagnostics rely on traceplots and effective-sample-size measures.

Empirical guidance for practitioners:

• Run sufficient MCMC iterations (e.g., $X_i$7 burn-in$X_i$8 posterior draws).
• Sensitivity analyses on frailty priors exhibit <7\% effect on MSMR.
• Bayesian marginal effects (BMEs) can be quantified by integrating over covariate subspaces, and the proportional-intensity assumption is testable via examination of time-varying BME estimates; time-constancy in log-BME suggests proportionality (Chen et al., 10 Jun 2026).

7. Extensions and Applications

The NHPP framework with SBART generalizes to handle interval- and right-censored survival data, spatial clustering, and unobserved covariates by integrating further latent-variable schemes (e.g., spatial CAR priors for cluster frailties) and multi-stage data augmentation (Ghosh et al., 2024). Smooth intensity estimation, uncertainty quantification, and predictive analysis are well supported. This approach is particularly suitable for biomedical and epidemiological recurrent event data, where subject-level heterogeneity and nonlinear dynamics cannot be parsimoniously specified in classical parametric or semiparametric survival models.

Summary Table: NHPP–SBART Framework Components

Component	Specification/Role	Reference
Intensity $X_i$9	$t$0	(Chen et al., 10 Jun 2026)
$t$1	SBART: $t$2 sum-of-soft-trees	(Chen et al., 10 Jun 2026)
Frailty $t$3	$t$4, hyperprior on $t$5	(Chen et al., 10 Jun 2026)
Data augmentation	Poisson thinning + latent Gaussian variables	(Chen et al., 10 Jun 2026)
Computation	Tree-wise MCMC, conjugate updates, C++ impl.	(Chen et al., 10 Jun 2026)
Empirical validation	Simulation, real-world hospitalizations	(Chen et al., 10 Jun 2026)

The NHPP framework with SBART establishes a scalable, nonparametric statistical model for recurrent events. Each modeling and inferential step, including data augmentation, hierarchical prior specification, and adaptation to nonlinearity, is formally documented in the cited literature (Chen et al., 10 Jun 2026).