  In this work we present the fundamental ideas of inference over paths, and
show how this formalism implies the continuity equation, which is central for
the derivation of the main partial differential equations that constitute
non-equilibrium statistical mechanics. Equations such as the Liouville
equation, Fokker-Planck equation, among others can be recovered as particular
cases of the continuity equation, under different probability fluxes. We derive
the continuity equation in its most general form through what we call the \bf
time-slicing equation, which lays down the procedure to go from the
representation in terms of a path probability functional $\rho[X()]$ to a
time-dependent probability density $\rho(x; t)$. The original probability
functional $\rho[X()]$ can in principle be constructed from different methods
of inference; in this work we sketch an application using the maximum path
entropy or maximum caliber principle.
