  Precise reconstruction of the cosmic microwave background lensing potential
can be achieved with deep polarization surveys by iteratively removing
lensing-induced $B$ modes. We introduce a lensing spectrum estimator and its
likelihood for such optimal iterative reconstruction. Our modelling share
similarities to the state-of-the-art likelihoods for quadratic estimator-based
(QE) lensing reconstruction. In particular, we generalize the $N_L^{(0)}$ and
$N_L^{(1)}$ lensing biases, and design a realization-dependent spectrum
debiaser, making this estimator robust to uncertainties in the data modelling.
We demonstrate unbiased recovery of the cosmology using map-based
reconstructions, focussing on lensing-only cosmological constraints and
neutrino mass measurement in combination with CMB spectra and acoustic
oscillation data. We find this spectrum estimator is essentially optimal and
with a diagonal covariance matrix. For a CMB-S4 survey, this likelihood can
double the constraints on the lensing amplitude compared to the QE on a wide
range of scales, while at the same time keeping numerical cost under control
and being robust to errors.
