  We present a novel technique for Cosmic Microwave Background (CMB) foreground
subtraction based on the framework of blind source separation. Inspired by
previous work incorporating local variation to Generalized Morphological
Component Analysis (GMCA), we introduce Hierarchical GMCA (HGMCA), a Bayesian
hierarchical graphical model for source separation. We test our method on
$N_{\rm side}=256$ simulated sky maps that include dust, synchrotron, free-free
and anomalous microwave emission, and show that HGMCA reduces foreground
contamination by $25\%$ over GMCA in both the regions included and excluded by
the Planck UT78 mask, decreases the error in the measurement of the CMB
temperature power spectrum to the $0.02-0.03\%$ level at $\ell>200$ (and
$<0.26\%$ for all $\ell$), and reduces correlation to all the foregrounds. We
find equivalent or improved performance when compared to state-of-the-art
Internal Linear Combination (ILC)-type algorithms on these simulations,
suggesting that HGMCA may be a competitive alternative to foreground separation
techniques previously applied to observed CMB data. Additionally, we show that
our performance does not suffer when we perturb model parameters or alter the
CMB realization, which suggests that our algorithm generalizes well beyond our
simplified simulations. Our results open a new avenue for constructing CMB maps
through Bayesian hierarchical analysis.
