Source
Schreiber (2000), “Measuring Information Transfer”, arXiv nlin/0001042v1 (Phys. Rev. Lett. 85, 461), LIT-1177, read in full in NOTE-940. The PRL version was not compared.
What was actually shown
The definition (Eq. 4). T_{J→I} = Σ p(i_{n+1}, i_n^{(k)}, j_n^{(l)}) log[p(i_{n+1} | i_n^{(k)}, j_n^{(l)}) / p(i_{n+1} | i_n^{(k)})], with i_n^{(k)} = (i_n, …, i_{n−k+1}) and j_n^{(l)} likewise. It is the Kullback entropy (Eq. 1) of the generalized Markov property p(i_{n+1} | i_n^{(k)}) = p(i_{n+1} | i_n^{(k)}, j_n^{(l)}), so it is nonnegative and zero exactly when that property holds. k and l are free; l = k and l = 1 are “most natural”.
Why it is directional. Mutual information is symmetric, and the asymmetry of conditional entropy is H_I − H_J, not flow. Eq. 4 conditions on the target’s own past, so correlation inherited from common history does not count.
Examples. In a tent-map lattice with binary partition, T in the coupling direction is of order ε² while delayed MI is zero to the same order; against the coupling T is zero. In an Ulam-map lattice T stays consistent with zero against the coupling through every bifurcation, and is zero in a period-two window where delayed MI is 1 bit.
What could have come out otherwise: in the Ulam lattice T against the coupling could have been nonzero in the windows where delayed MI was nearly symmetric. It was not.
What this does not say
- That positive T is information moving from J to I. The Letter calls it “transfer” and “exchange” but defines nothing beyond Eq. 4, a conditional dependence. Whether that is a flow is CLAIM-335’s question.
- That unobserved common drivers are excluded. Only a known driver Z, conditioned on, is (p. 2). The heart–breath asymmetry is left open to a common trigger by the author.
- That T is interventional. Eq. 4 is observational; when it bears on interventions is THEORY-704’s and CLAIM-333’s question.
- Which k and l to use. Results at k = l = 1 are not shown to survive longer histories.
- Anything about estimation error. No bias, variance or null distribution is given; THEORY-641 supplies a χ² null under a parametric model.
- The tent-map constant. The reading’s evaluation of Eq. 4 gives α²ε²/(2 ln 2), half the Letter’s α²ε²/ln 2. With α fitted, nothing qualitative changes.
Connections
- THEORY-641 and THEORY-700 are later results about this quantity: a likelihood-ratio limit in general, half the Granger statistic for Gaussians.
- THEORY-275 uses a delayed variant, I(y_{t+τ}; x_t | y_t), which is Eq. 4 at k = l = 1 only for τ = 1.
- THEORY-568 counts T as one summand of a Shannon invariant’s numerator.
- TERM-057 is the record’s own statement of the term.