  One of the major targets for next-generation cosmic microwave background
(CMB) experiments is the detection of the primordial B-mode signal. Planning is
under way for Stage-IV experiments that are projected to have instrumental
noise small enough to make lensing and foregrounds the dominant source of
uncertainty for estimating the tensor-to-scalar ratio $r$ from polarization
maps. This makes delensing a crucial part of future CMB polarization science.
In this paper we present a likelihood method for estimating the
tensor-to-scalar ratio $r$ from CMB polarization observations, which combines
the benefits of a full-scale likelihood approach with the tractability of the
quadratic delensing technique. This method is a pixel space, all order
likelihood analysis of the quadratic delensed B modes, and it essentially
builds upon the quadratic delenser by taking into account all order lensing and
pixel space anomalies. Its tractability relies on a crucial factorization of
the pixel space covariance matrix of the polarization observations which allows
one to compute the full Gaussian approximate likelihood profile, as a function
of $r$, at the same computational cost of a single likelihood evaluation.
