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 N( ABPE) algorithm with an ingenious update rule of the auxiliary phase designed via entropy-based sampling. Unlike adaptive Bayesian phase estimation (ABPE) algorithms, the auxiliary phase in our algorithm is design de by a pre-established iteration rule with simple statistical analysis of a small batch of data from a pre-estimation process, instead of complex calculations in every update trails. During the pre-estimation process the most informative data can be selected via entropy-based sampling, which guides one to perform the Bayesian phase estimation with much less measurement times. As the measurement times for the same amount of Bayesian updates is significantly reduced, our NABPE algorithm via entropy b- ased sampling can work as efficient as ABPE algorithms and shares the advantages (such as wide dynamic range and perfect noise robustness) of NABPE algorithms. Our algorithm is of promising applications in various practical quantum sensors such as at mo ic clocks and quantum magnetometers.