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We introduce an ambidextrous view of stochastic dynamical systems, comparingtheir forward-time and reverse-time representations and then integrating theminto a single time-symmetric representation. The perspective is usefultheoretically, computationally, and conceptually. Mathematically, we prove thatthe excess entropy--a familiar measure of organization in complex systems--isthe mutual information not only between the past and future, but also betweenthe predictive and retrodictive causal states. Practically, we exploit theconnection between prediction and retrodiction to directly calculate the excessentropy. Conceptually, these lead one to discover new system invariants forstochastic dynamical systems: crypticity (information accessibility) and causalirreversibility. Ultimately, we introduce a time-symmetric representation thatunifies all these quantities, compressing the two directional representationsinto one. The resulting compression offers a new conception of the amount ofinformation stored in the present.

Prediction, Retrodiction, and The Amount of Information Stored in the Present