  Bayesian estimation approaches, which are capable of combining the
information of experimental data from different likelihood functions to achieve
high precisions, have been widely used in phase estimation via introducing a
controllable auxiliary phase. Here, we present a non-adaptive Bayesian phase
estimation (BPE) algorithms with an ingenious update rule of the auxiliary
phase designed via active learning. Unlike adaptive BPE algorithms, the
auxiliary phase in our algorithm is determined by a pre-established update rule
with simple statistical analysis of a small batch of data, instead of complex
calculations in every update trails. As the number of measurements for a same
amount of Bayesian updates is significantly reduced via active learning, our
algorithm can work as efficient as adaptive ones and shares the advantages
(such as wide dynamic range and perfect noise robustness) of non-adaptive ones.
Our algorithm is of promising applications in various practical quantum sensors
such as atomic clocks and quantum magnetometers.
