Source
Williams & Beer (2011), “Generalized Measures of Information Transfer”, arXiv 1102.1507v1, LIT-1176, read in full in NOTE-942.
What was actually shown
The identity (Eqs. 4–8). If I(X; Y) = Π({Y}) + Π({Y}{Z}) and I(X; Y, Z) = Π({Y}) + Π({Z}) + Π({Y}{Z}) + Π({Y, Z}), then I(X; Y | Z) = Π({Y}) + Π({Y, Z}). So T_{Y→X} = SITE + SDTE and I(X_{t+1}; Y_t) = SITE + redundancy.
Control (Theorem 1). Perfect open-loop controllability iff perfect controllability with SDTE = 0, under I_min.
Examples, recomputed exactly. In the two-process family T = 1 − ½h(d), pure SITE at d = 0 and pure SDTE (a controlled-NOT) at d = 1; at d = ½, SITE = 0.189 and SDTE = 0.311 bits. In the two-source family the corners give 1 bit of unique, redundant and synergistic transfer.
Physiology. Heart→breath transfer in a sleep-apnoea recording is state-dependent, with SITE consistent with zero both ways. Not checked.
What could have come out otherwise: the apnoea data could have shown SITE, which would have let TE and TDMI share a component. They share none there.
What this does not say
- What the sizes of SITE and SDTE are under another redundancy measure. Only the split is measure-free. The numbers away from the corners are I_min’s.
- That SDTE is information flow. It is the synergistic term THEORY-707 says is not localizable to the source. “Transfer” is the authors’ word.
- That SITE and SDTE grade open- and closed-loop influence in general. Theorem 1 is about perfect controllability.
- That the apnoea result is significant. No test is reported.
Connections
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Under Ince’s I_ccs (THEORY-714) the coupled pair is again unchanged. Harder, Salge and Polani’s bystander example gives a third value at d = 0.75, 0.392/0.157, and a negative state-independent part at d = 0.25 (NOTE-968).
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The coupled pair’s split is the same under Harder, Salge and Polani’s I_red (THEORY-708) and Bertschinger et al.’s unique information (THEORY-705), at every d checked. Elsewhere the split depends on the measure: in Harder, Salge and Polani’s bystander example at d = 0.75 it is 0/0.549 under I_min and 0.411/0.138 under I_red (NOTE-944).
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THEORY-707 uses this decomposition against the flow reading.
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THEORY-568’s averages need only the consistency equations, the same property the split here needs.
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THEORY-497 records that ΦID verdicts can turn on the redundancy function; so can SITE and SDTE.
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TERM-057 states transfer entropy in the record’s words.