Finite-state Markov Chains Obey Benford’s Law
Bibliographic record
Abstract
A sequence of real numbers ([Formula: see text]) is Benford if the significands, i.e., the fraction parts in the floating-point representation of ([Formula: see text]), are distributed logarithmically. Similarly, a discrete-time irreducible and aperiodic finite-state Markov chain with transition probability matrix [Formula: see text] and limiting matrix [Formula: see text] is Benford if every component of both sequences of matrices ([Formula: see text]) and ([Formula: see text]) is Benford or eventually zero. Using recent tools that established Benford behavior for finite-dimensional linear maps, via the classical theories of uniform distribution modulo 1 and Perron–Frobenius, this paper derives a simple sufficient condition (“nonresonance”) guaranteeing that [Formula: see text], or the Markov chain associated with it, is Benford. This result in turn is used to show that almost all Markov chains are Benford, in the sense that if the transition probability matrix is chosen in an absolutely continuous manner, then the resulting Markov chain is Benford with probability one. Concrete examples illustrate the various cases that arise, and the theory is complemented with simulations and potential applications.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.003 | 0.015 |
| Meta-epidemiology (narrow) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.002 | 0.004 |
| Scholarly communication | 0.003 | 0.004 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.003 | 0.003 |
| Insufficient payload (model declined to judge) | 0.006 | 0.002 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".