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In this paper, we present the first study that compares different models ofBayesian Neural Networks (BNNs) to predict the posterior distribution of thecosmological parameters directly from the Cosmic Microwave Backgroundtemperature and polarization maps. We focus our analysis on four differentmethods to sample the weights of the network during training: Dropout,DropConnect, Reparameterization Trick (RT), and Flipout. We find out thatFlipout outperforms all other methods regardless of the architecture used, andprovides tighter constraints for the cosmological parameters. Moreover wecompare with MCMC posterior analysis obtaining comparable error correlationamong parameters, with BNNs being orders of magnitude faster in inference,although less accurate. Thanks to the speed of the inference process with BNNs,the posterior distribution, outcome of the neural network, can be used as theinitial proposal for the Markov Chain. We show that this combined approachincreases the acceptance rate in the Metropolis-Hasting algorithm andaccelerates the convergence of the MCMC, while reaching the same finalaccuracy. In the second part of the paper, we present a guide to the trainingand calibration of a successful multi-channel BNN for the CMB temperature andpolarization map. We show how tuning the regularization parameter for thestandard deviation of the approximate posterior on the weights in Flipout andRT we can produce unbiased and reliable uncertainty estimates, i.e., theregularizer acts like a hyperparameter analogous to the dropout rate inDropout. Finally, we show how polarization, when combined with the temperaturein a unique multi-channel tensor fed to a single BNN, helps to breakdegeneracies among parameters and provides stringent constraints.

Parameters Estimation for the Cosmic Microwave Background with Bayesian Neural Networks