abstract: We address Bayesian estimation of a phase shift imposed to a qubit when non-dissipative phase noise affects its propagation. The comparison with the ultimate quantum limit to precision, the quantum Cramer-Rao bound, is given. In our analysis we also take into account the biased nature of the Bayes estimator in the non-asymptotic regime. The experimental demonstration of our scheme relates on a setup based on a KDP crystal, that allows both to manipulate the optical qubit polarization and to simulate the noise affecting the propagation. An adaptive method to always achieve the optimal estimation, i.e., the Cramer-Rao bound, is also analyzed in some detail.