  The inference of Markov models from data on stochastic dynamical trajectories
over the large time-window $T$ is revisited via the Large Deviations at Level
2.5 for the time-empirical density and the time-empirical flows. The goal is to
obtain the large deviations properties for the probability distribution of the
inferred Markov parameters in order to characterize their possible fluctuations
around the true Markov parameters for large $T$. The explicit rate functions
are given for several settings, namely discrete-time Markov chains,
continuous-time Markov jump processes, and diffusion processes in dimension
$d$. Applications to various models of random walks in disordered media are
described, where the goal is to infer the quenched disordered variables
defining a given disordered sample.
