Addressing prior dependence in hierarchical Bayesian modeling for PTA data analysis II: Noise and SGWB inference through parameter decorrelation

TL;DR

The paper tackles prior dependence in hierarchical Bayesian PTA analyses by decorrelating hyperparameters from per-pulsar noise parameters through an orthogonal reparameterization learned with Normalizing Flows, coupled with flow-guided nested sampling via i-nessai. Using a minimal 3-pulsar setup, it shows that introducing hyperpriors on noise components reduces red-noise–SGWB degeneracy and that decorrelation yields near-independence between hyperparameters and physical parameters while tightening red-noise inferences. The SGWB estimates remain largely unchanged in this small-array context, highlighting that the method primarily stabilizes noise parameter inference rather than the common SGWB signal in limited data. The approach promises reduced prior sensitivity and more interpretable hierarchical PTA models, with potential for substantial gains in larger arrays and real datasets for robust SGWB characterization.

Abstract

Pulsar Timing Arrays provide a powerful framework to measure low-frequency gravitational waves, but accuracy and robustness of the results are challenged by complex noise processes that must be accurately modeled. Standard PTA analyses assign fixed uniform noise priors to each pulsar, an approach that can introduce systematic biases when combining the array. To overcome this limitation, we adopt a hierarchical Bayesian modeling strategy in which noise priors are parametrized by higher-level hyperparameters. We further address the challenge posed by the correlations between hyperparameters and physical noise parameters, focusing on those describing red noise and dispersion measure variations. To decorrelate these quantities, we introduce an orthogonal reparametrization of the hierarchical model implemented with Normalizing Flows. We also employ i-nessai, a flow-guided nested sampler, to efficiently explore the resulting higher-dimensional parameter space. We apply our method to a minimal 3-pulsar case study, performing a simultaneous inference of noise and SGWB parameters. Despite the limited dataset, the results consistently show that the hierarchical treatment constrains the noise parameters more tightly and partially alleviates the red-noise-SGWB degeneracy, while the orthogonal reparametrization further enhances parameter independence without affecting the correlations intrinsic to the power-law modeling of the physical processes involved.

Figures (7)

• Figure 1: Architecture of the i-nessai hierarchical model, shown in both the non-reparametrized (hyperparameters $\bm{\Lambda}$) and reparametrized (hyperparameters $\tilde{\bm{\Lambda}}$) forms. This implementation handles standard hierarchical Bayesian inference in the two situations: the first with explicit uniform conditional priors $\pi(\bm{\vartheta}|\bm{\Lambda})$ and Gaussian or uniform hyperpriors $\pi'(\bm{\Lambda})$; the second with a pair of Normalizing Flows: the push-formward PF-NF encoding $\pi'(\tilde{\bm{\Lambda}})$ and the pull-back PB-CNF encoding $\pi(\bm{\vartheta}|\tilde{\bm{\Lambda}})$. The likelihood is evaluated via Enterprise and priors and likelihood feed the sampler i-nessai in parallel. The output are the posterior samples.
• Figure 2: Corner plot showing the 2D-joint posterior distributions of the noise and SGWB parameters for pulsar J1744$-$1134 under fixed, non-hierarchical priors. The two-dimensional contours display the characteristic anticorrelation between the logarithmic amplitude and spectral index typical of the power-law modeling of both DM variations and SGWB. In contrast, the intrinsic red noise is almost unconstrained in this configuration: with only three pulsars in the dataset, the data provide little sensitivity to individual red-noise parameters, resulting in broad, nearly flat posteriors that indicate the dominance of prior information. These results provide the reference case for assessing the impact of the hierarchical modeling and reparametrization presented in Section \ref{['subsec:method']}.
• Figure 3: Marginal posterior distributions for the noise and SGWB parameters of pulsar J1744$-$1134 obtained with fixed uniform priors. The red vertical lines indicate the injected values. The sharp upper-edge peak of the SGWB amplitude posterior primarily reflects the prior-bound effect, but is also reinforced by the limited number of pulsars and by the intrinsic red-noise–SGWB degeneracy. This behaviour is particularly evident in small arrays, where the Hellings–Downs correlation drives the sampler to attribute most of the frequency power to the common SGWB component.
• Figure 4: Corner plot of the noise and SGWB parameters (left) and independence score metric for the correlations between the physical and hyperparameters for pulsar J1744$-$1134 with a hierarchical treatement of the noise
• Figure 5: Marginal posterior distributions for the noise and SGWB parameters of pulsar J1744$-$1134 obtained with Gaussian hyperpriors on the red-noise and DM-variation priors. The red vertical lines mark the injected values. Compared to the fixed-prior case, the red-noise posteriors are more constrained, while the SGWB parameters remain largely unchanged, indicating that in this small-array configuration the hierarchical treatment mainly regularizes the noise components.