Fig. 3: Information diagrams for the past-dependent information measures.
From: Duality between predictability and reconstructability in complex systems
In a, we show the information diagram of the random variable triplet \(({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{past}}}}}}}}},{{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{future}}}}}}}}},G)\), where Xpast represents the past states, \({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{future}}}}}}}}}\), the future states, and G, the structure of the system. The quantities of interest are \(I({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{future}}}}}}}}};G| {{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{past}}}}}}}}})\) indicated by the green set, \(H({{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{future}}}}}}}}}| {{{{{{{{\bf{X}}}}}}}}}_{{{{{{{{\rm{past}}}}}}}}})\) shown by the union of the pink and green sets, and H(G∣Xpast), represented by the union of the blue and green sets. b, c Two extreme scenarios where the length of the past τ is small and large, which illustrates how the different information measures change with τ.
