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

Approximate Likelihood Approaches for Detecting the Influence of Primordial Gravitational Waves in Cosmic Microwave Background Polarization