Extracting important parameters from dynamical systems models through coarse-graining

Known microscopic details often motivate models with large numbers of parameters. However, not all parameter combinations are relevant at large length-scales of observation. Some may affect coarse-grained system behavior sensitively, while others may have no discernible effect. Here we utilize the Fisher Information Metric formalism to identify which parameter combinations influence observables even for coarse-grained data. We first derive a general method for calculating this metric from a model whose data has been coarse-grained, and apply this to models coarse-grained by sparse-sampling observables in time. We make the resulting Fisher Information reparameterization invariant by transforming to a basis that measures how coarse-graining reduces observability. We then use this procedure to explicitly calculate the temporally coarse-grained Fisher Information Metric for several stochastic differential equation models. The expansion of the reparameterization invariant Fisher spectrum after coarse-graining separates relevant parameter combinations from irrelevant ones. We then draw concrete parallels between our formalism which uses coarse-graining of the Fisher Information, and the Renormalization Group framework.