  The existence of a cosmic background of primordial gravitational waves (PGWB)
is a robust prediction of inflationary cosmology, but it has so far evaded
discovery. The most promising avenue of its detection is via measurements of
Cosmic Microwave Background (CMB) $B$-polarization. However, this is not
straightforward due to (a) the fact that CMB maps are distorted by
gravitational lensing and (b) the high-dimensional nature of CMB data, which
renders likelihood-based analysis methods computationally extremely expensive.
In this paper, we introduce an efficient likelihood-free, end-to-end inference
method to directly infer the posterior distribution of the tensor-to-scalar
ratio $r$ from lensed maps of the Stokes $Q$ and $U$ polarization parameters.
Our method employs a generative model to delense the maps and utilizes the
Approximate Bayesian Computation (ABC) algorithm to sample $r$. We demonstrate
that our method yields unbiased estimates of $r$ with well-calibrated
uncertainty quantification.
