Parameter estimation for the cosmic microwave background with Bayesian neural networks

In this paper, we present the first study that compares different models of Bayesian neural networks (BNNs) to predict the posterior distribution of the cosmological parameters directly from the cosmic microwave background (CMB) temperature and polarization maps. We focus our analysis on four different methods to sample the weights of the network during training: Dropout, DropConnect, Reparameterization Trick (RT), and Flipout. We find that Flipout outperforms all other methods regardless of the architecture used, and provides tighter constraints for the cosmological parameters. Moreover, we compare our results with a Markov chain Monte Carlo (MCMC) posterior analysis and obtain comparable error correlations among parameters, with BNNs that are orders of magnitude faster in inference, albeit less accurate. Thanks to the speed of the inference process with BNNs, the posterior distribution-the outcome of the neural network-can be used as the initial proposal for the Markov chain. We show that this combined approach increases the acceptance rate in the Metropolis-Hasting algorithm and accelerates the convergence of the MCMC, while reaching the same final accuracy. In the second part of the paper, we present a guide to the training and calibration of a successful multichannel BNN for the CMB temperature and polarization map. We show how tuning the regularization parameter for the standard deviation of the approximate posterior on the weights in Flipout and RT can produce unbiased and reliable uncertainty estimates, i.e., the regularizer acts like a hyperparameter analogous to the dropout rate in Dropout. The best performances are nevertheless achieved with a more convenient method, in which the network parameters are kept free during training to achieve the best uncalibrated performances, and the confidence intervals are calibrated in a subsequent phase. Additionally, we describe existing strategies for calibrating the networks and propose new ones. Finally, we show how polarization, when combined with the temperature in a unique multichannel tensor fed to a single BNN, helps to break degeneracies among parameters and provides stringent constraints. The results reported in this paper can be extended to other cosmological data sets in order to capture features that can be extracted directly from the raw data, such as non-Gaussianity or foreground emissions.