  In this study, we address the central issue of statistical inference for
Markov jump processes using discrete time observations. The primary problem at
hand is to accurately estimate the infinitesimal generator of a Markov jump
process, a critical task in various applications. To tackle this problem, we
begin by reviewing established methods for generating sample paths from a
Markov jump process conditioned to endpoints, known as Markov bridges.
Additionally, we introduce a novel algorithm grounded in the concept of
time-reversal, which serves as our main contribution. Our proposed method is
then employed to estimate the infinitesimal generator of a Markov jump process.
To achieve this, we use a combination of Markov Chain Monte Carlo techniques
and the Monte Carlo Expectation-Maximization algorithm. The results obtained
from our approach demonstrate its effectiveness in providing accurate parameter
estimates. To assess the efficacy of our proposed method, we conduct a
comprehensive comparative analysis with existing techniques (Bisection,
Uniformization, Direct, Rejection, and Modified Rejection), taking into
consideration both speed and accuracy. Notably, our method stands out as the
fastest among the alternatives while maintaining high levels of precision.
