Elusive present: Hidden past and future dependency and why we build models

Modeling a temporal process as if it is Markovian assumes that the present encodes all of a process's history. When this occurs, the present captures all of the dependency between past and future. We recently showed that if one randomly samples in the space of structured processes, this is almost never the case. So, how does the Markov failure come about? That is, how do individual measurements fail to encode the past? and How many are needed to capture dependencies between the past and future? Here, we investigate how much information can be shared between the past and the future but not reflected in the present. We quantify this elusive information, give explicit calculational methods, and outline the consequences, the most important of which is that when the present hides past-future correlation or dependency we must move beyond sequence-based statistics and build state-based models.

Figure 1

The process information diagram, which places the present in its temporal context: the past ($X_{:0}$) and the future ($X_{\ell :}$) partition the present ($X_{0:\ell }$) into four components with quantities $r_{\mu }^{\ell }$ and $q_{\mu }^{\ell }$ and two with $b_{\mu }^{\ell }$. Notably, the component ${\sigma }_{\mu }^{\ell }$, quantifying the hidden dependency shared by the past and the future, lies outside of the present and so is not part of it.

Figure 2

Alternative decompositions of the present information $H\left[X_{0:\ell }\right]$. (a) Decomposition due to the past, (b) decomposition due to the past and the future.

Figure 3

The mutual information $\mathbf{I}[X_{:0}:X_{0:\ell }]$ between the past and the present (shaded) is equivalent to the mutual information $\mathbf{I}[{\mathcal{S}}_0^{+}:X_{0:\ell }]$ between the forward causal state and the present.

Figure 4

Random variable lattice illustrating the relationship among forward causal states ${\mathcal{S}}_t^{+}$, observed symbols $X_t$, and reverse causal states ${\mathcal{S}}_t^{-}$. Variables in the distribution $Pr({\mathcal{S}}_{-1}^{+},X_{-1:2},{\mathcal{S}}_2^{-})$ are highlighted. In particular, the elusivity ${\sigma }_{\mu }^3$ is the mutual information between the two shaded cells (${\mathcal{S}}_{-1}^{+}$ and ${\mathcal{S}}_2^{-}$) conditioned on the hatched cells ($X_{-1:2}=X_{-1}{X}_0{X}_1$).

Figure 5

The several faces of the Golden Mean process. (a) Forward $\epsilon$-machine, (b) reverse $\epsilon$-machine, (c) bimachine. Each representation consists of states (labeled circles) and transitions (arrows) labeled “symbol:probability”. The bimachine is effectively the Cartesian product of the forward and reverse machines, though constructed here for forward-time generation, and therefore is generically nonunifilar. For example, state B:C means that the machine is in the superposition of forward state B and reverse state C. Going forward (rightward in Fig. 4) we know that state B must output a 0 and transition to state A. Being in reverse state C we must have transitioned there on a 0 coming from either state C or state D. Thus, we have transitions from B:C to both A:C and A:D on a 0.

Figure 6

$\epsilon$-Machines for the example processes: (a) The Even process, (b) the Noisy Period Three (NP3) process, (c) the Noisy Random Phase-slip (NRPS) process.

Figure 7

Information measures as a function of the present's length $\ell$. Since the examples are stationary, finite processes, both ${\sigma }_{\mu }^{\ell }$ and $q_{\mu }^{\ell }$ converge to 0 with increasing $\ell$ and $b_{\mu }^{\ell }$ converges to a constant value of $\mathbf{E}$. And so, the growth of $H\left(\ell \right)$ is entirely captured in $r_{\mu }^{\ell }$, and it grows linearly with $\ell$.

Figure 8

${\sigma }_{\mu }^{\ell }$ (solid line) and its asymptote (dashed line) for (a) the Nemo process with $K=0.6$ and $\alpha =0.64$ and (b) the even process with $K=1$ and $\alpha =0.71$.

Figure 9

The Nemo process.

Figure 10

Persistent mutual information $PMI\left(\infty \right)$, elusivity ${\sigma }_{\mu }^{\infty }$, and multivariate mutual information $q_{\mu }^{\infty }$ of the Logistic Map symbolic dynamics as a function of the map control parameter $r$. Recall that the symbolic dynamics does not see period doubling until $r$ is above the appearance of the associated superstable periodic orbit. This discrepancy in the appearance of periodicity as a function of $r$ does not occur when the map is chaotic. (Compare Fig. 1 in Ref. [15].)