  With a single training image and using wavelet phase harmonic augmentation,
we present polarized Cosmic Microwave Background (CMB) foreground
marginalization in a high-dimensional likelihood-free (Bayesian) framework. We
demonstrate robust foreground removal using only a single frequency of
simulated data for a BICEP-like sky patch. Using Moment Networks we estimate
the pixel-level posterior probability for the underlying {E,B} signal and
validate the statistical model with a quantile-type test using the estimated
marginal posterior moments. The Moment Networks use a hierarchy of U-Net
convolutional neural networks. This work validates such an approach in the most
difficult limiting case: pixel-level, noise-free, highly non-Gaussian dust
foregrounds with a single training image at a single frequency. For a real CMB
experiment, a small number of representative sky patches would provide the
training data required for full cosmological inference. These results enable
robust likelihood-free, simulation-based parameter and model inference for
primordial B-mode detection using observed CMB polarization data.
