# Bayesian Inference of Primordial Magnetic Field Parameters from CMB with Spherical Graph Neural Networks

arXiv: https://arxiv.org/abs/2510.20795 · HTML: https://mlsanity.com/paper/2510.20795/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 16 of its 40 references that are in the map.
- First posted on arXiv 23 Oct 2025.

## Builds on (most cited first, 16 of 16)

- [On Calibration of Modern Neural Networks](https://mlsanity.com/paper/1706.04599/) (2017)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Accurate Uncertainties for Deep Learning Using Calibrated Regression](https://mlsanity.com/paper/1807.00263/) (2018)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [Flipout: Efficient Pseudo-Independent Weight Perturbations on Mini-Batches](https://mlsanity.com/paper/1803.04386/) (2018)
- [Graph Neural Networks in Network Neuroscience](https://mlsanity.com/paper/2106.03535/) (2021)
- [DeepSphere: Efficient spherical Convolutional Neural Network with HEALPix sampling for cosmological applications](https://mlsanity.com/paper/1810.12186/) (2018)
- [Bayesian Layers: A Module for Neural Network Uncertainty](https://mlsanity.com/paper/1812.03973/) (2018)
- [Bayesian Learning for Neural Networks: an algorithmic survey](https://mlsanity.com/paper/2211.11865/) (2022)
- [DeepCMB: Lensing Reconstruction of the Cosmic Microwave Background with Deep Neural Networks](https://mlsanity.com/paper/1810.01483/) (2018)
- [Parametric and Multivariate Uncertainty Calibration for Regression and Object Detection](https://mlsanity.com/paper/2207.01242/) (2022)
- [Machine Learning and Cosmology](https://mlsanity.com/paper/2203.08056/) (2022)
- [Probabilistic Calibration by Design for Neural Network Regression](https://mlsanity.com/paper/2403.11964/) (2024)
- [Accelerating MCMC algorithms through Bayesian Deep Networks](https://mlsanity.com/paper/2011.14276/) (2020)
- [Bayesian deep learning for cosmic volumes with modified gravity](https://mlsanity.com/paper/2309.00612/) (2023)

## Abstract (arXiv)

Deep learning has emerged as a transformative methodology in modern cosmology, providing powerful tools to extract meaningful physical information from complex astronomical datasets. This paper implements a novel Bayesian graph deep learning framework for estimating key cosmological parameters in a primordial magnetic field (PMF) cosmology directly from simulated Cosmic Microwave Background (CMB) maps. Our methodology utilizes DeepSphere, a spherical convolutional neural network architecture specifically designed to respect the spherical geometry of CMB data through HEALPix pixelization. To advance beyond deterministic point estimates and enable robust uncertainty quantification, we integrate Bayesian Neural Networks (BNNs) into the framework, capturing aleatoric and epistemic uncertainties that reflect the model confidence in its predictions. The proposed approach demonstrates exceptional performance, achieving R2 scores exceeding 0.89 for the magnetic parameter estimation. We further obtain well-calibrated uncertainty estimates through post-hoc training techniques including Variance Scaling and GPNormal. This integrated DeepSphere-BNNs framework not only delivers accurate parameter estimation from CMB maps with PMF contributions but also provides reliable uncertainty quantification, providing the necessary tools for robust cosmological inference in the era of precision cosmology.