  Milestoning is an efficient method for rare event kinetics calculation using
short trajectory parallelization. Mean first passage time (MFPT) is the key
kinetic output of Milestoning, whose accuracy crucially depends the initial
distribution of the short trajectory ensemble. The true initial distribution,
i.e., first hitting point distribution (FHPD), has no analytic expression in
the general case. Here, we introduce two algorithms, local passage time
weighted Milestoning (LPT-M) and Bayesian inference Milestoning (BI-M), to
accurately and efficiently approximate FHPD for systems at equilibrium
condition. Starting from sampling Boltzmann distribution on milestones, we
calculate the proper weighting factor for the short trajectory ensemble. The
methods are tested on two model examples for illustration purpose. Both methods
improve significantly over the widely used classical Milestoning method in
terms of the accuracy of MFPT. In particular, BI-M covers the directional
Milestoning method as a special case in the deterministic Hamiltonian dynamics.
LPT-M is especially advantageous in terms of computational costs and robustness
with respect to the increasing number of intermediate milestones. Furthermore,
a locally iterative correction algorithm for non-equilibrium stationary FHPD is
developed for exact MFPT calculation, which can be combined with LPT-M/BI-M and
is much cheaper than the exact Milestoning method.
