Lionel Barnett, Terry Bossomaier (2012), Physical Review Letters 109, 138105 (2012; online 2012-09-28) — ARXIV-1205.6339
Key takeaways
- For a broad class of Markov predictive models, the log-likelihood-ratio statistic for “zero transfer entropy” is a consistent estimator of the transfer entropy itself, with a standard asymptotic χ² null distribution. This generalizes the Gaussian equivalence of transfer entropy and Granger causality.
Seeded from the abstract and a skim, before the reading in NOTE-053. What follows is what the work says about itself.
Transfer entropy measures time-directed information flow between processes and is widely used in neuroscience, ecology, climatology and econometrics, but it is hard to estimate. The authors show that, for a wide class of parametric predictive models, the likelihood-ratio test statistic for the null of zero transfer entropy, suitably normalized, converges to the transfer entropy. For finite-state Markov chains no explicit model is needed, because the plug-in estimator is already of this form. In general the estimator is asymptotically χ²-distributed, which permits significance tests. The result extends the known Gaussian equivalence of transfer entropy and Granger causality, and ties Schreiber-style information transfer to Wiener–Granger causality.
Standing in the record
Filed from the survey of 2026-09-25 of work the anthology set aside as out of scope (tier C): 380 seconds of active reading over 3 sessions in the papers-feed tracker. It was filed Deferred, unread. NOTE-053 is the close reading of 2026-09-25, and it placed the work: Active — a short, correct letter and the standard citation for reading transfer entropy as a log-likelihood ratio. The consistency proof is informal, and the χ² results are cited to Wald (1943) rather than proved, but the identity itself is exact and still the one to cite.
When seeded, before the reading in NOTE-053, the entry gave a priority for a deeper reading: low — A short, self-contained letter whose skim already captures the theorem. The owner’s time is moderate (t=380 s). It carries no ML recommendation.
What the seeding asked a deeper reading to check (NOTE-053 is that reading):
- It is a clean bridge between an information-theoretic quantity and standard likelihood inference. That is useful background for any causal or directed-information analysis of time series, including analysis of learned systems.
- A deeper reading should check the ergodicity condition (eq. 10) and how slowly the non-central χ² converges in practice (the authors flag it). Also check how the result compares with the later literature on transfer-entropy estimator bias.
Access when seeded: I read the arXiv abstract page (v1 29 May 2012, v3 27 Jul 2012) and the full v3 PDF (10 pp., about 6 pp. of text). Crossref confirms the PRL DOI and its online date of 2012-09-28. The arXiv title capitalizes “Log-likelihood” and PRL writes “Log-Likelihood”.