Ultimate limit on learning non-Markovian behavior: Fisher information rate and excess information

We address the fundamental limits of learning unknown parameters of any stochastic process from time-series data, and discover exact closed-form expressions for how optimal inference scales with observation length. Given a parametrized class of candidate models, the Fisher information of observed sequence probabilities lower-bounds the variance in model estimation from finite data. As sequence-length increases, the minimal variance scales as the square inverse of the length -- with constant coefficient given by the information rate. We discover a simple closed-form expression for this information rate, even in the case of infinite Markov order. We furthermore obtain the exact analytic lower bound on model variance from the observation-induced metadynamic among belief states. We discover ephemeral, exponential, and more general modes of convergence to the asymptotic information rate. Surprisingly, this myopic information rate converges to the asymptotic Fisher information rate with exactly the same relaxation timescales that appear in the myopic entropy rate as it converges to the Shannon entropy rate for the process. We illustrate these results with a sequence of examples that highlight qualitatively distinct features of stochastic processes that shape optimal learning.