Definition
Schreiber’s Eq. 4 (NOTE-940), in the record’s notation:
T_{Y→X}^{(k,l)} = Σ p(x_{t+1}, x_t^{(k)}, y_t^{(l)}) log [p(x_{t+1} | x_t^{(k)}, y_t^{(l)}) / p(x_{t+1} | x_t^{(k)})] = I(X_{t+1}; Y_t^{(l)} | X_t
with x_t^{(k)} = (x_t, …, x_{t−k+1}). It is the Kullback entropy of the assumption that Y’s past leaves X’s transition law unchanged, and is zero exactly when it does. A conditional form adds the past of further series Z to both conditionals. Stated with it, always:
- k and l. Schreiber leaves both free and uses k = l = 1 in his examples. Lizier et al. argue k → ∞ is the right limit, since a short destination history credits the source with what the destination’s own past would have predicted (THEORY-702). NOTE-053 uses k-histories. NOTE-788’s TE_{x→y}(τ) = I(y_{t+τ}; x_t | y_t) is the k = l = 1 case at τ = 1. For τ > 1 it conditions on y_t, not y_{t+τ−1}, and is a delayed variant Schreiber does not define.
- The conditioning set. Apparent transfer entropy conditions on the target’s past only. Complete transfer entropy also conditions on the target’s other causal parents, which must then be known (Lizier et al., Eq. 14).
- Average or local. The transfer entropy is the average of a local quantity, log of the same ratio at one time step, which can be negative where the source misleads (THEORY-702).
- The estimator, for any number reported. Under a well-specified parametric model it is a likelihood-ratio statistic with a χ² null (THEORY-641). Nearest-neighbour estimates inherit the bias of Kraskov et al.’s construction (THEORY-709), and that χ² null does not apply to them.
What it must not be confused with
- Information flow, if flow means information in the target attributable to the source alone. Transfer entropy is unique plus synergistic information (CLAIM-335, THEORY-707, THEORY-712). The words “transfer” and “flow” in Schreiber and in Williams and Beer are the authors’ usage. The record reads them as directed predictive information.
- Time-delayed mutual information, I(X_{t+1}; Y_t). It drops the target’s past, so it counts redundancy and misses synergy, where transfer entropy does the reverse (THEORY-712).
- Directed information (Massey), Σ I(X^n; Y_n | Y^{n−1}). It conditions on the current source value. Full-history transfer entropy is its lagged part; the instantaneous remainder is what transfer entropy misses (THEORY-711).
- Interventional information flow (Ay and Polani), the same formula with interventional probabilities. It coincides with transfer entropy only when conditioning on the pasts is as good as setting them and every configuration has positive probability (THEORY-710).
- Granger causality (TERM-058). For jointly Gaussian processes, Geweke’s Granger causality is twice the transfer entropy (THEORY-700). What CLAIM-333 says of Granger causality and interventions holds for transfer entropy too.