  Using the probabilistic language of conditional expectations we reformulate
the force matching method for coarse-graining of molecular systems as a
projection on spaces of coarse observables. A practical outcome of this
probabilistic description is the link of the force matching method with
thermodynamic integration. This connection provides a way to systematically
construct a local mean force in order to optimally approximate the potential of
mean force through force matching. We introduce a generalized force matching
condition for the local mean force in the sense that allows the approximation
of the potential of mean force under both linear and non-linear coarse graining
mappings (e.g., reaction coordinates, end-to-end length of chains).
Furthermore, we study the equivalence of force matching with relative entropy
minimization which we derive for general non-linear coarse graining maps. We
present in detail the generalized force matching condition through applications
to specific examples in molecular systems.
