PUREPath-B: A Tessellated Bayesian Model for Recovering CMB B-modes over Large Angular Scales of the Sky

We introduce a comprehensive, custom-developed neural network, the PUREPath-B, that yields a posterior predictive distribution of Cosmic Microwave Background (CMB) B-mode signal conditioned on the foreground contaminated CMB data and informed by the training dataset. Our network employs nested probabilistic multi-modal U-Net framework, enhanced with probabilistic ResNets at skip connections and seamlessly integrates Bayesian statistics and variational methods to minimize the foreground and noise contaminations. During training, the initial prior distribution over network parameters evolves into approximate posterior distributions through Bayesian inference, constrained by the training data. From the approximate joint full posterior of the model parameters, our network infers a predictive CMB posterior during inference and yields summary statistics such as predictive mean, variance of the cleaned map. The predictive standard deviation provides an interpretable measure of per-pixel uncertainty in the predicted mean CMB map. For loss function, we use a linear combination of KL-Divergence loss and weighted MAE-which ensures that maps with higher amplitudes do not dominate the loss disproportionately. Furthermore, the results from the cosmological parameter estimation using the cleaned B-mode power spectrum, along with its error estimates demonstrates our network minimizes the foreground contaminations effectively, enabling accurate recovery of tensor-to-scalar ratio and lensing amplitude.