  We study a Data-Driven approach to inference in physical systems in a
measure-theoretic framework. The systems under consideration are characterized
by two measures defined over the phase space: i) A physical likelihood measure
expressing the likelihood that a state of the system be admissible, in the
sense of satisfying all governing physical laws; ii) A material likelihood
measure expressing the likelihood that a local state of the material be
observed in the laboratory. We assume deterministic loading, which means that
the first measure is supported on a linear subspace. We additionally assume
that the second measure is only known approximately through a sequence of
empirical (discrete) measures. We develop a method for the quantitative
analysis of convergence based on the flat metric and obtain error bounds both
for annealing and the discretization or sampling procedure, leading to the
determination of appropriate quantitative annealing rates. Finally, we provide
an example illustrating the application of the theory to transportation
networks.
