  This paper explores methods for constructing low multipole temperature and
polarisation likelihoods from maps of the cosmic microwave background
anisotropies that have complex noise properties and partial sky coverage. We
use the Planck 2018 High Frequency Instrument (HFI) maps and the updated SRoll2
maps (Delouis et al. 2019) to test our methods. We present three likelihood
approximations based on quadratic cross spectrum estimators: (i) a variant of
the simulation-based likelihood (SimBaL) techniques used in the Planck legacy
papers to produce a low multipole EE likelihood; (ii) a semi-analytical
likelihood approximation (momento) based on the principle of maximum entropy;
(iii) a density-estimation "likelihood-free" scheme (DELFI). Approaches (ii)
and (iii) can be generalised to produce low multipole joint
temperature-polarisation (TTTEEE) likelihoods. We present extensive tests of
these methods on simulations with realistic correlated noise. We then analyse
the Planck data and confirm the robustness of our method and likelihoods on
multiple inter- and intra-frequency detector set combinations of SRoll2 maps.
The three likelihood techniques give consistent results and support a low value
of the optical depth to reoinization, tau, from the HFI. Our best estimate of
tau comes from combining the low multipole SRoll2 momento (TTTEEE) likelihood
with the CamSpec high multipole likelihood and is tau = 0.0627+0.0053-0.0059.
This is consistent with the value of tau reported by Pagano et al. (2020),
though slightly higher by approximately 0.5 sigma.
