Direct estimates of irreversibility from time series

Gianluca Teza; Trevor GrandPre; William Bialek

REVIEW 3 major objections 5 minor 1 cited by

Direct estimates of irreversibility from time series

T0 review · 3 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A model-free estimator of forward-reverse trajectory divergence, corrected for finite-sample bias, recovers zero irreversibility in equilibrium and detects a nonzero arrow of time in single retinal neurons.

desk verdict A clean bias-correction recipe for plug-in irreversibility estimates, with a provocative retina result that outruns the paper's own validation. read the letter →

arxiv 2412.19772 v1 pith:RKPG6XOE submitted 2024-12-27 cond-mat.stat-mech physics.bio-phq-bio.NCstat.AP

Trevor GrandPre , Gianluca Teza , William Bialek This is my paper · ORCID

classification cond-mat.stat-mechphysics.bio-phq-bio.NCstat.AP PACS 05.70.Ln

keywords Kullback-Leiblerdivergencearrowoftimeirreversibilityfinite-samplebiasretinalganglioncellsnon-Markoviandynamicsentropyproductionmodel-freeestimation

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper tries to establish that the arrow of time can be read directly from observed time series, with no model of the underlying dynamics. It quantifies irreversibility as the Kullback-Leibler divergence ($D_{KL}$), a measure of how different two probability distributions are, between the distribution of trajectory windows and their time reverses. The central technical point is that the naive plug-in estimate of this divergence is systematically biased upward by finite sample size, with a computable $1/N$ plus $1/N^2$ correction, so extrapolating to infinite data removes the bias. Calibrated on simulations, the corrected estimator returns $D_{KL}=0$ for a detailed-balance system and the right entropy-production rate for a driven cycle; applied to salamander retinal neurons it finds nonzero irreversibility in single cells over 200 ms windows, implying strongly non-Markovian dynamics. If this works, it gives a reliable way to detect time-asymmetry in systems where no heat flow or thermodynamic flux is directly measurable.

What carries the argument

The machinery is the plug-in estimator of Eq. (9) joined to the bias expansion of Eq. (10). The central object is $D_{KL}$, the Kullback-Leibler divergence between the probability of a discretized trajectory word $W$ and the probability of its time-reversed partner $\tilde W$; the estimator counts how much more often forward words appear than reversed words. Because empirical counts fluctuate, the plug-in estimate is biased upward, with a leading term $A/N$ whose coefficient is $A=\frac{1}{2}[\Omega'+\sum_W P(W)/P(\tilde W)]$. Extrapolating across sample sizes removes the leading bias, and using the asymptotic form $D_{KL}(T)\to\sigma T+\sigma_1$ extracts a rate $\sigma$ from finite windows.

What would settle it

Generate a long stationary, time-reversible but non-Markovian time series with known zero irreversibility, for example a Gaussian process with symmetric autocorrelation, and apply the estimator to 200 ms windows: if the extrapolated $D_{KL}$ is systematically nonzero across many realizations the bias expansion is missing a term, while if it is zero within error the correction works as claimed.

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Extended reading notes

Core claim

The central claim is that a model-free, direct estimator of the forward-reverse trajectory divergence can be made quantitatively reliable by correcting its finite-sample bias. The plug-in estimator is $\hat{D}_N(T)=\sum_W P_N(W)\ln[P_N(W)/P_N(\tilde W)]$, and the paper derives the bias expansion $\langle \hat{D}_N(T)\rangle = D_{KL}(T)+A/N+B/N^2+\cdots$ with $A=\frac{1}{2}[\Omega'+\sum_W P(W)/P(\tilde W)]$, then extrapolates to $N\to\infty$. On a simulated two-state Markov chain obeying detailed balance this recovers $D_{KL}=0$ within a standard deviation, and on a driven three-state cycle the extrapolated $D_{KL}(T)/T$ converges to the known entropy-production rate $\sigma\approx20.7\mathrm{s}^{-1}$ when the time bin is small enough. On salamander retinal data, the extrapolated $D_{KL}(T=200\mathrm{ms})$ is nonzero for almost every single neuron, vanishes after shuffling the spike trains, and grows supralinearly with $T$ on the 40-200 ms scale.

Load-bearing premise

The method assumes the observed trajectory windows are independent draws from a single fixed distribution, so that the finite-sample bias has exactly the $1/N$ plus $1/N^2$ form that the extrapolation removes; no test of this assumption is reported for the retina spike trains.

Editorial extensions

If this is right

A nonzero extrapolated $D_{KL}$ from a stationary time series is evidence that the underlying dynamics are not time-reversal invariant, without needing a physical model of the system.
For Markovian dynamics the method recovers the thermodynamic entropy-production rate from trajectory data alone, matching the analytic rate when the discretization bin is small enough.
Because any binary Markov model obeys detailed balance, the observed nonzero $D_{KL}$ in single retinal neurons identifies the non-Markovian character of the spike train.
The supralinear growth of $D_{KL}(T)$ means that evidence for the arrow of time accumulates synergistically across successive brief windows rather than additively.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural next test is to push the same estimator to longer windows, where the supralinear growth should cross over to a linear $D_{KL}(T)\sim\sigma T$ if a finite entropy-production rate exists; locating that crossover would give a rate for retinal spike trains.
Because the $1/N$ coefficient involves ratios $P(W)/P(\tilde W)$, data sets in which many time-reversed words are never observed may need a regularized estimator, and sensitivity of the extrapolation to that regulator would reveal when the bias expansion has broken down.
Applied to pairs or populations of simultaneously recorded neurons, the same method could test whether the arrow-of-time evidence is enhanced by correlations between cells, rather than only by temporal correlations within each cell.
The method should transfer to other high-dimensional time series with a well-defined forward-reverse pairing, such as gene expression or weather records, where model-based entropy-production estimates are unavailable.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Colleague,

The useful takeaway: this is a short Letter that imports the well-known entropy-estimation trick—expand the finite-sample bias in powers of 1/N and extrapolate—to the DKL between forward and reverse trajectories. The new piece is the explicit bias formula, Eqs. (10)-(11), and the application to single retinal ganglion cells. The synthetic checks are genuinely clean: the estimator returns zero for a two-state equilibrium system and matches the analytic entropy-production rate for a three-state cycle. That part is credible.

The problem is in the step from synthetic to real data. Eq. (9) is undefined whenever a word is observed but its time reverse is not; the paper never says how zero counts are handled. Any fix—pseudocount, dropping those words—changes the bias, and the 1/N expansion has to be reworked accordingly. That is a concrete omission.

The larger issue is the independence assumption behind Eq. (6) and hence Eq. (10). For the retina data, the 200 ms words are cut from a continuous spike train; they are not independent samples, and the paper does not test whether the 1/N + 1/N^2 form actually holds in that regime. The shuffle control in Fig. 3 only checks that the estimator is unbiased when the data have been decorrelated; it says nothing about a time-reversible but correlated process. So the nonzero single-neuron DKL could in principle be an artifact of the extrapolation, and the paper doesn't rule that out.

I think the authors know the limitations; the text says 'there is no magic' and admits the extrapolation relies on the same logic used for entropy estimation. But for a method paper, the missing validation on a time-reversible correlated null is a real gap, not a nitpick.

Still, the paper deserves a serious referee. The bias expansion is a useful concrete result, the synthetic validation is honest, and the application is interesting. I'd send it out with a request that the authors specify the regularization, and that they add a surrogate test that preserves correlations while enforcing time-reversal symmetry.

Recommendation: conditional accept, with those two additions. It's a solid Letter, not a game-changer.

Referee Report

3 major / 5 minor

Summary. The paper proposes a model-free estimator of the Kullback-Leibler divergence between the distributions of forward and time-reversed trajectories, based on plug-in frequency estimates and a finite-sample bias correction. The authors derive a bias expansion (Eqs. 10-11), extrapolate to infinite sample size, validate the method on a two-state Markov chain (recovering D_KL = 0) and a three-state cycle (recovering the analytic entropy-production rate), and apply it to spike trains from salamander retinal ganglion cells, reporting nonzero D_KL for single neurons and supralinear growth with window length.

Significance. If the estimator is valid in the regimes where it is applied, this is a significant methodological contribution: it provides a direct, dynamics-agnostic route to quantifying irreversibility from time series, with clean synthetic validation and a novel neural-data application. The paper also gives a concrete formula for the leading finite-sample bias, which is useful beyond the specific extrapolation procedure. However, the strength of the retina claim depends on assumptions about sample independence and the handling of unobserved words, and these assumptions are not checked; the method as presented is therefore not yet established in the regime of its headline application.

major comments (3)

[Eq. (9) and Section 'We start by discretizing'] The plug-in estimator in Eq. (9) is undefined whenever a word W is observed but its time reverse ~W is not: the term ln[PN(W)/PN(~W)] diverges. The manuscript never states how zero reverse counts are handled (pseudocounts, omission, or otherwise), and any such regularization changes the bias and invalidates the unmodified expansion in Eqs. (10)-(11). This is not a technicality for the retina application, where rare words are common; the authors must specify the procedure used and account for its effect on the bias.
[Eq. (10) and Fig. 3] The bias expansion in Eq. (10) is derived from Eq. (6), which assumes that the N word samples are independent draws from the true distribution. In the retina experiment, the 200 ms words are cut from a continuous spike train, and consecutive words are likely overlapping and correlated; the paper reports no test of independence and no check that the 1/N + 1/N^2 extrapolation is valid for such data. The shuffle control in Fig. 3 breaks temporal correlations, so it only demonstrates unbiasedness for independent samples; it does not test the null hypothesis of a time-reversible but correlated process. If the effective sample size is much smaller than N, the extrapolation can leave a residual bias that masquerades as genuine irreversibility, directly undermining the central claim about single-neuron D_KL.
[Eqs. (10)-(11)] The derivation of the bias expansion is only sketched ('following the same path as for the entropy'), and the coefficient A in Eq. (11) is asserted without a full derivation or a reference. Since the entire extrapolation procedure rests on this expansion, the authors should provide a complete derivation (or a supplementary appendix) that covers the treatment of palindromic words, the definition of Omega', and the conditions under which the 1/N^2 term has the stated form.

minor comments (5)

[Eq. (8)] The standard Miller-Madow correction for the plug-in entropy estimator is (Omega - 1)/(2N), not Omega/(2N); if the authors intend the large-Omega approximation, they should state this explicitly.
[Abstract and Section 'In summary'] The term 'model-free' is used for a method that assumes a particular parametric bias expansion and sample independence; consider softening this wording to 'assumption-light' or 'dynamics-agnostic'.
[Fig. 1 caption] The caption contains a typo: '1=N' should be '1/N'.
[Fig. 3 inset] The inset shows D_KL(T) normalized by D_KL(T=10 Delta_tau) and averaged over ten neurons; please clarify how the normalization is performed and whether the error bars include the across-cell variance.
[Retina application] The text does not report the number of spike trains, the total recording duration, or the number of words used per cell; these details are necessary to assess the reliability of the extrapolation in Fig. 3.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the bias-corrected D_KL estimator is validated against analytic models and the extrapolated target is not used to set fit parameters.

full rationale

The central derivation is self-contained. The plug-in estimator in Eq (9) is a direct definition-based estimator of Eq (2); the bias expansion in Eq (10) is derived from the sampling covariance in Eq (6), with the leading coefficient in Eq (11) computed from the underlying distribution. The extrapolation fits the nuisance coefficients A and B to the observed 1/N dependence and reads off the intercept D_KL; the target is not used to constrain the fit, so the recovery of D_KL = 0 for a two-state Markov chain and the analytic slope for the three-state cycle are genuine external checks, not consequences of the fitting procedure. The retina measurements are an application of the calibrated estimator. The paper does cite prior work by the same group (e.g., Refs. [34,35,37,40,41,49,52-55]), but these citations are not load-bearing for the derivation: the entropy-extrapolation strategy is also attributed to independent sources [33,36], and no uniqueness theorem or ansatz is imported from the authors' own papers to force the choice of estimator. Concerns that the independence assumption behind Eq (6) may fail for correlated 200 ms neural windows, or that zero reverse counts are not discussed, are correctness/robustness issues rather than circular reductions; they do not make the estimate equal to its input by construction.

Assumptions & free parameters 3 free parameters · 6 assumptions · 0 invented entities

The central claim rests on a statistical extrapolation procedure rather than on new physical entities. The estimator has no invented particles or forces, but it relies on several unproven domain assumptions about sample independence, the validity of the 1/N expansion, and the behavior of the plug-in estimator when reverse words are unseen. The nuisance coefficients A and B are fit to the data, so the reported DKL values depend on the stability of that fit.

free parameters (3)

A = not reported
Bias coefficient of 1/N in Eq (10); the extrapolation estimates DKL as the intercept while A is a free fit parameter, and the final estimates depend on this fit.
B = not reported
Second-order bias coefficient of 1/N^2 in Eq (10); fit alongside A, with no constraints or stability analysis reported.
sigma_1 = not reported
Additive constant in Eq (14) used in Fig 2 to fit DKL(T) versus 1/T and extract the entropy production rate; not central to the retina findings but part of the validation analysis.

assumptions (6)

domain assumption The plug-in estimator bias has a regular expansion in powers of 1/N (Eq 10) with finite coefficients A and B.
Needed for the extrapolation to N→∞; the derivation is only sketched and assumes independent samples and that all relevant words are observed.
domain assumption The N word samples are independent draws from P(W).
Used to derive Eq (6) and the bias expansion; not verified for overlapping windows in retina data.
domain assumption Correlations between consecutive words diminish subextensively with increasing word length, so longer words approach independence.
Invoked to justify the extrapolation strategy; may fail for non-Markovian processes with long memory.
domain assumption At large T, DKL(T) ≈ σT + σ1 (Eq 14).
Used to interpret Fig 2 and extract entropy production rate; standard for finite-correlation-time processes but not proven for all systems.
domain assumption Shuffling the time axis produces a valid null with DKL=0.
Used as a control in Fig 3; reasonable but the precise shuffling procedure is not described.
domain assumption The support of the reverse distribution includes all observed forward words, so plug-in ratios are finite in the N→∞ limit.
Required for Eq (9) to be well-defined; not satisfied for finite samples without regularization.

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Pith. "Pith review of Direct estimates of irreversibility from time series." pith.science (2026). https://pith.science/paper/RKPG6XOE

@misc{pith2026241219772,
  author       = {Pith},
  title        = {Pith review of: Direct estimates of irreversibility from time series},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RKPG6XOE}},
  note         = {Machine review of arXiv:2412.19772}
}

abstract

The arrow of time can be quantified through the Kullback-Leibler divergence ($D_{KL}$) between the distributions of forward and reverse trajectories in a system. Many approaches to estimate this rely on specific models, but the use of incorrect models can introduce uncontrolled errors. Here, we describe a model-free method that uses trajectory data directly to estimate the evidence for irreversibility over finite windows of time. To do this we build on previous work to identify and correct for errors that arise from limited sample size. Importantly, our approach accurately recovers $D_{KL} = 0$ in systems that adhere to detailed balance, and the correct nonzero $D_{KL}$ for data generated by well understood models of nonequilibrium systems. We apply our method to trajectories of neural activity in the retina as it responds to naturalistic inputs, and find evidence of irreversibility in single neurons, emphasizing the non-Markovian character of these data. These results open new avenues for investigating how the brain represents the arrow of time.

Figures

Figures reproduced from arXiv: 2412.19772 by the authors.

**Figure 2.** Irreversibility in a three state system. Extrapolated [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** The arrow of time in the activity of single neurons. [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Forward citations

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G. Teza, S. Iubini, M. Baiesi, A. L. Stella, and C. Van- derzande, Rate dependence of current and fluctuations in jump models with negative differential mobility, Phys- ica A: Statistical Mechanics and its Applications, 552, 6 123176 (2020)

2020
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Angeli, S

L. Angeli, S. Grosskinsky, A. M. Johansen, and A. Piz- zoferrato, Rare event simulation for stochastic dynamics in continuous time. Journal of Statistical Physics 176, 1185 (2019)

2019

Pith tools

Reviewed August 10, 2026 · model on record in the stance chip above.

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I. A. Mart ´ ınez, G. Bisker, J. M. Horowitz, and J. M. R. Parrondo. Inferring broken detailed balance in the absence of observable currents. Nature Communications 10(1), 3542 (2019)

work page 2019

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Van der Meer, B

J. Van der Meer, B. Ertel, and U. Seifert, Thermody- namic inference in partially accessible Markov networks: a unifying perspective from transition-based waiting time distributions. Physical Review X12(3), 031025 (2022)

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P. E. Harunari, A. Dutta, M. Polettini, and ´E. Rold´ an, What to learn from a few visible transitions’ statistics? Physical Review X12(4), 041026 (2022)

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Ehrich, Tightest bound on hidden entropy production from partially observed dynamics

J. Ehrich, Tightest bound on hidden entropy production from partially observed dynamics. Journal of Statisti- cal Mechanics: Theory and Experiment2021(8), 083214 (2021)

work page 2021

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Ghosal and G

A. Ghosal and G. Bisker, Inferring entropy production rate from partially observed Langevin dynamics under coarse-graining. Physical Chemistry Chemical Physics 24(39), 24021–24031 (2022)

work page 2022

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Pietzonka and F

P. Pietzonka and F. Coghi, Thermodynamic cost for pre- cision of general counting observables. Physical Review E 109(6), 064128 (2024)

work page 2024

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Baiesi, T

M. Baiesi, T. Nishiyama, and G. Falasco, Effective esti- mation of entropy production with lacking data. Com- munications Physics 7(1), 264 (2024)

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Ghosal and G

A. Ghosal and G. Bisker, Entropy production rates for different notions of partial information. Journal of Physics D: Applied Physics56(25), 254001 (2023)

work page 2023