Learning Informed Prior Distributions with Normalizing Flows for Bayesian Analysis

Abstract

We investigate the use of normalizing flow (NF) models as flexible priors in Bayesian inference via Markov Chain Monte Carlo (MCMC) sampling for iterative Bayesian calibration. Trained on posteriors from previous analyses, these models can be used as informative priors that capture non-trivial distributions and correlations in subsequent inference tasks. We compare different training strategies and loss functions, finding that training based on Kullback-Leibler (KL) divergence and unsupervised learning consistently yield the most accurate reproductions of reference distributions. We apply such a sequential Bayesian workflow to a high-energy nuclear physics problem; MCMC with NF-based priors reproduces the results of one-shot joint inference well, provided the target distributions are unimodal. In cases with pronounced multi-modality or dataset tension, distortions may arise, underscoring the need for caution in multi-stage Bayesian inference. A comparison between the pocoMC MCMC sampler and the standard emcee sampler further demonstrates the importance of advanced and robust algorithms for exploring the posterior space. Overall, our results establish NF-based priors as a practical and efficient tool for sequential Bayesian inference in high-dimensional parameter spaces.

Figures (7)

• Figure 1: Comparison of the target distribution (green) with samples from the NF model (orange, dotted) for the posterior constrained with the $\gamma+p$ dataset Mantysaari:2025ltq. The NF model utilizes the KL loss function, a batch size of 5000, 6 layers, and a learning rate of $1\times 10^{-3}$. Contour lines indicate the $1\sigma$, $2\sigma$, and $3\sigma$ boundaries.
• Figure 2: Comparison of the target distribution (green) with samples from the NF model (orange, dotted) for the posterior constrained with the $\gamma+\mathrm{Pb}$ dataset Mantysaari:2025ltq. The NF model utilizes the KL loss function, a training batch size of 1000, 12 layers, and a learning rate of $1 \times 10^{-3}$. Contour lines indicate the $1\sigma$, $2\sigma$, and $3\sigma$ boundaries.
• Figure 3: Multi-stage Bayesian inference starting from the posterior constrained with the $\gamma+p$ data, then inference with the $\gamma+\mathrm{Pb}$ data. Full lines (green) indicate the joint inference, dotted lines (orange) the first-stage posterior, and dashed lines (purple) the second-stage posterior. Contours show $1\sigma$, $2\sigma$, and $3\sigma$ boundaries.
• Figure 4: Multi-stage Bayesian inference starting from the posterior constrained by the $\gamma+\mathrm{Pb}$ data, then inference with the $\gamma+p$ data. Full lines (green) indicate the joint inference, dotted lines (orange) the first-stage posterior, and dashed lines (purple) the second-stage posterior. Contours show $1\sigma$, $2\sigma$, and $3\sigma$ boundaries.
• Figure 5: Same inference as Fig. \ref{['fig:MCMC_eP_prior']}, but replacing the second-stage pocoMC sampler with emcee. Contours show $1\sigma$, $2\sigma$, and $3\sigma$ boundaries.