Addressing prior dependence in hierarchical Bayesian modeling for PTA data analysis I: Methodology and implementation

TL;DR

This paper tackles prior dependence in hierarchical Bayesian PTA analyses by introducing an NF-based orthogonal reparameterization that decorrelates hyperparameters from physical PTA parameters. It combines Push-forward and Pull-backward Normalizing Flows with a flow-guided nested sampler (i-nessai) to efficiently explore complex, high-dimensional posteriors within the Enterprise PTA framework. Validation on a single-pulsar DR2-like dataset demonstrates that the method can reduce prior-driven variability and accelerate inference, while also identifying residual correlations that motivate future improvements, such as more flexible priors and physics-informed flows. The approach offers a principled pathway to robust, scalable hierarchical PTA inference with potential impact on SGWB detection and pulsar-noise characterization.

Abstract

Complex inference tasks, such as those encountered in Pulsar Timing Array (PTA) data analysis, rely on Bayesian frameworks. The high-dimensional parameter space and the strong interdependencies among astrophysical, pulsar noise, and nuisance parameters introduce significant challenges for efficient learning and robust inference. These challenges are emblematic of broader issues in decision science, where model over-parameterization and prior sensitivity can compromise both computational tractability and the reliability of the results. We address these issues in the framework of hierarchical Bayesian modeling by introducing a reparameterization strategy. Our approach employs Normalizing Flows (NFs) to decorrelate the parameters governing hierarchical priors from those of astrophysical interest. The use of NF-based mappings provides both the flexibility to realize the reparametrization and the tractability to preserve proper probability densities. We further adopt i-nessai, a flow-guided nested sampler, to accelerate exploration of complex posteriors. This unified use of NFs improves statistical robustness and computational efficiency, providing a principled methodology for addressing hierarchical Bayesian inference in PTA analysis.

Figures (7)

• Figure 1: Pipeline for hierarchical decorrelation: sample ($\bm{\theta},{\bm\Lambda}$), project to $\tilde{\bm{\Lambda}}$, learn $\pi\left(\bm{\Lambda}\right)$ and $\pi\left(\bm{\theta}|\bm{\Lambda}\right)$ with NFs, then infer with i-nessai.
• Figure 2: Training (blue) and validation (orange) losses for both unconditional (\ref{['fig:unconditional']}) and conditional networks (\ref{['fig:conditional']}) on a linear scale. The vertical red dashed line indicates the selected early stopping epoch. Overall, the losses converge to low and stable values, confirming efficient and robust training.
• Figure 3: Real data PDF (blue) vs PF-NF samples (orange) for eight $\tilde{\Lambda}$ dimensions; close agreement shows the model reproduces the target distribution.
• Figure 4: Real data PDF (blue) vs PB-CNF samples (orange) for four $\vartheta$ dimensions; close agreement indicates the model reproduces the target distribution.
• Figure 5: Single-pulsar RN and DM posteriors under two hyperpriors: uniform (\ref{['fig:uni_gamma']}, \ref{['fig:uni_A']}) vs Gaussian (\ref{['fig:gauss_gamma']}, \ref{['fig:gauss_A']}); red lines mark injected values. The hyperprior choice materially affects the inferred parameters (Enterprise likelihood with i-nessai).