  We address the problem of estimating the transmissivity of the pure-loss
channel from the Bayesian point of view, i.e., we consider that some prior
probability distribution function (PDF) on the unknown variable is available
and we employ methods to compute the Bayesian minimum mean square error (MMSE).
Specifically, we consider two prior PDFs: the two-point and the beta
distributions. By fixing the input mean photon number to an integer, for the
two-point PDF we prove analytically that the optimal state is the Fock state
and the optimal measurement is photon-counting, while for the beta PDF our
numerical investigation provides evidence on the optimality of the Fock state
and photon-counting. Moreover, we investigate the situation where the input
mean photon number is any (non-negative) real number. For said case, we
conjecture the form of the optimal input states and we study the performance of
photon-counting, which is a sub-optimal yet practical measurement. Our methods
can be applied for any prior PDF. We emphasize that we compute the MMSE instead
of Bayesian lower bounds on the mean square error based on the Fisherian
approach.
