  Bayesian estimation is a powerful theoretical paradigm for the operation of
quantum sensors. However, the Bayesian method for statistical inference
generally suffers from demanding calibration requirements that have so far
restricted its use to proof-of-principle experiments. In this theoretical
study, we formulate parameter estimation as a classification task and use
artificial neural networks to efficiently perform Bayesian estimation. We show
that the network's posterior distribution is centered at the true (unknown)
value of the parameter within an uncertainty given by the inverse Fisher
information, representing the ultimate sensitivity limit for the given
apparatus. When only a limited number of calibration measurements are
available, our machine-learning based procedure outperforms standard
calibration methods. Thus, our work paves the way for Bayesian quantum sensors
which can benefit from efficient optimization methods, such as in adaptive
schemes, and take advantage of complex non-classical states. These capabilities
can significantly enhance the sensitivity of future devices.
