  In this paper we develop a new data-driven closure approximation method to
compute the statistical properties of quantities of interest in
high-dimensional stochastic dynamical systems. The new method relies on
estimating conditional expectations from sample paths or experimental data, and
it is independent of the dimension of the underlying phase space. We also
address the important question of whether enough useful data is being injected
into the reduced-order model governing the quantity of interest. To this end,
we develop a new paradigm to measure the information content of data based on
the numerical solution of hyperbolic systems of equations. The effectiveness of
the proposed new methods is demonstrated in applications to nonlinear dynamical
systems and models of systems biology evolving from random initial states.
