Improving Bayesian inference in PTA data analysis: importance nested sampling with Normalizing Flows

TL;DR

This work develops and benchmarks flow-based nested sampling (i-nessai) integrated into the Enterprise PTA framework to accelerate Bayesian PTA analyses. By training Normalizing Flows to represent likelihood-constrained priors at each nested level and forming a meta-proposal, it achieves accurate posteriors and reliable evidence with substantially reduced runtimes compared to PTMCMC. Extensive diagnostics demonstrate stable convergence, calibrated uncertainty estimation, and robust parameter recovery across noise-only and SGWB-included models for simulated PTA datasets. The results indicate that flow-based nested sampling can dramatically improve scalability and reliability of PTA inferences, enabling efficient model comparisons and hierarchical analyses in current and next-generation PTA datasets.

Abstract

We present a detailed study of Bayesian inference workflows for pulsar timing array data with a focus on enhancing efficiency, robustness and speed through the use of normalizing flow-based nested sampling. Building on the Enterprise framework, we integrate the i-nessai sampler and benchmark its performance on realistic, simulated datasets. We analyze its computational scaling and stability, and show that it achieves accurate posteriors and reliable evidence estimates with substantially reduced runtime, by up to three orders of magnitude depending on the dataset configuration, with respect to conventional single-core parallel-tempering MCMC analyses. These results highlight the potential of flow-based nested sampling to accelerate PTA analyses while preserving the quality of the inference.

Figures (7)

• Figure 1: State plot from i-nessai for the inference of the 5 noise parameters (amplitude and spectral index for both intrinsic red noise and DM variation and amplitude for white noise (EFAC)) per pulsar for a simulated dataset comprising 10 pulsars. 50 parameters in total. For a discussion of each panel see Subsection \ref{['sec:state']}. Flow configuration: 6 blocks, 4 layers, 64 neurons per layer, single thread, pool size 12. Total wall-time 6h 52min (82.4 CPU h). Run performed on the HPC cluster at Università degli Studi di Milano-Bicocca.
• Figure 2: Corner plot showing the posterior distributions and correlations between i-nessai internal parameters for the optimized run corresponding to the state plot in Figure \ref{['fig:state-best']}) for 10 pulsars. Flow configuration: 6 blocks, 4 layers, 64 neurons per layer, single thread, pool size 12. The diagonal panels display the marginalized one-dimensional distributions for each parameter, while the off-diagonal panels show the two-dimensional joint distributions with 1$\sigma$ and 2$\sigma$ contours.
• Figure 3: Stability analysis of i-nessai across 10 independent runs with identical model configurations but different random seeds for a 3-pulsar simulated dataset. (a) Violin plot showing the distribution of log-evidence values, with mean $\log Z = 53032.18 \pm 0.0070$. The narrow distribution demonstrates excellent reproducibility across stochastic initializations. (b) Comparison between the empirical standard deviation of log-evidence across runs and the mean reported uncertainty, yielding a ratio of 0.007 that indicates well-calibrated internal error estimates. (c) Wall-time distribution with mean $6.68 \pm 0.15$ minutes, showing consistent computational performance with minimal variation. (d) Recovery of the injected red noise amplitude parameter ($\log_{10}A_{\text{rn}} = -13.155$, shown as horizontal dashed line) for three pulsars, with error bars representing 68% credible intervals. The coefficient of variation for log-evidence (0.01%) and wall-time (2.2%) are both exceptionally low, indicating that single runs provide representative results without the need for extensive replication. Run performed on a 20-core Intel i9 laptop (32 GB RAM). Total wall-time 1h 2min 19s (6.23 CPU h)
• Figure 4: Corner plot showing the posterior distributions and correlations for the noise parameters of pulsar J0030+0451 in the noise-only model. The diagonal panels display the marginalized one-dimensional posterior distributions for each parameter, while the off-diagonal panels show the two-dimensional joint distributions with 1$\sigma$ and 2$\sigma$ contours. The five parameters are: red noise amplitude ($\log_{10} A_{\rm RN}$) and spectral index ($\gamma_{\rm RN}$), dispersion measure variation amplitude ($\log_{10} A_{\rm DM}$) and spectral index ($\gamma_{\rm DM}$), and white noise amplitude (EFAC). The posteriors are obtained using i-nessai with a simulated dataset of 10 pulsars from EPTA DR2new, with flow configuration: 6 blocks, 4 layers, 64 neurons per layer.
• Figure 5: Trace plot showing the evolution of noise parameters for pulsar J0030+0451 as a function of nested sampling iteration. Each panel displays one parameter, with points colored by their normalized importance weight (darker colors indicate higher weights). The horizontal dashed red line indicates the posterior mean. The concentration of high-weight samples in the later iterations demonstrates the efficiency of the nested sampling algorithm in progressively constraining the parameter space.