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Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior

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Pith's one-line read Tsallis and Sharma–Mittal entropies of the Poisson distribution always increase with intensity λ, but two generalized Rényi entropies can decrease over intervals of λ—an 'anomalous' non-monotonicity.

desk verdict Solid, modest paper: the Tsallis/Sharma-Mittal monotonicity results are clean, the anomaly claim is plausible, but two advertised bounds in Theorem 4.1 have a wrong exponential factor and need correction. read the letter →

arxiv 2411.16913 v1 pith:HIVGFUFE submitted 2024-11-25 math.PR cs.ITmath.IT

Dmitri Finkelshtein , Anatoliy Malyarenko , Yuliya Mishura , Kostiantyn Ralchenko This is my paper · ORCID
classification math.PRcs.ITmath.IT MSC 94A1760E05
keywords ShannonentropyRényiTsallisSharma–MittalgeneralizedPoissondistributionmonotonicityasymptoticbounds
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 asks which entropy measures of a Poisson distribution with intensity λ keep their intuitive monotone behavior as λ grows. It proves that the Tsallis entropy H_T(α,λ) and the Sharma–Mittal entropy H_SM(α,β,λ) are strictly increasing in λ for every admissible parameter value, joining the already known monotonicity of the Shannon and Rényi entropies. In contrast, it proves that the one-parameter generalized Rényi entropy H_GR(α,λ) can decrease as λ increases: for each α in a small interval near 0, H_GR(α,λ) is decreasing in a neighborhood of λ=1, and numerical evidence indicates similar non-monotone behavior for large α. The two-parameter generalized Rényi entropy H_GR(α,β,λ) inherits this anomalous decrease when the parameters are close to a region where H_GR(α,·) decreases. The paper also gives leading-order asymptotics as λ→∞ and two-sided bounds for the entropies, the Shannon case matching the known (1/2)log(2πλ) growth.

What carries the argument

The load-bearing object is ψ(α,λ)=Σ_{i≥0} p_i(λ)^α=$e^{{-αλ}}$Σ_{i≥0} $λ^{{iα}}$/(i!)^α. Every one of the six entropies is a functional of ψ and its logarithmic derivative: H_R=(1/(1−α))log ψ, H_T=(ψ−1)/(1−α), H_GR(α,β)=(log ψ(α,·)−log ψ(β,·))/(β−α), H_SM=($ψ^{{(1−β)/(1−α)}}$−1)/(1−β), and H_GR(α,·)=−∂_α log ψ. Since ψ is increasing in λ for 0<α<1 and decreasing for α>1, the signs in the definitions force Shannon, Rényi, Tsallis, and Sharma–Mittal entropies to increase in λ. The anomalous cases come from the mixed derivative: ∂_λ H_GR(α,λ)=−∂_α∂_λ log ψ(α,λ), evaluated at λ=1 through ρ(α)=α((Σ_{i≥1} i/(i!)^α)/(Σ_{i≥0}1/(i!)^α)−1); when ρ increases in α, the generalized Rényi entropy initially falls as λ rises.

What would settle it

Compute ρ(α)=α((Σ_{i≥1} i/(i!)^α)/(Σ_{i≥0}1/(i!)^α)−1) and ρ′(α) for α=$10^{{-k}}$, k=3,...,12, using the series or high-precision quadrature, and check whether ρ′(α)>0 on an entire interval near 0. Proposition 5.4 predicts such an interval; if ρ′≤0 for all sufficiently small α, the claimed decreasing behavior of H_GR(α,λ) near λ=1 would not follow from the given argument. Conversely, isolating an α with ∂_λ H_GR(α,λ)|_{λ=1}<0 would confirm the anomaly directly.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central result is a classification of 'normal' versus 'anomalous' behavior for six entropy families on Poisson probabilities p_i(λ)=$e^{{-λ}}$λ^i/i!. Proposition 5.1 (with earlier work) establishes that Shannon, Rényi, Tsallis, and Sharma–Mittal entropies are strictly increasing functions of λ for all admissible parameter values. Proposition 5.4 establishes the existence of an interval J⊂(0,1) near 0 such that for every α∈J, the generalized Rényi entropy H_GR(α,λ)=−∂_α log ψ(α,λ) is decreasing in λ in a neighborhood of λ=1, so monotonicity fails even though ρ(α)=∂_λ log ψ|_{λ=1} is positive on (0,1). Proposition 5.8 transfers this non-monotonicity to the two-parameter entropy H_GR(α,β,λ) for β near such an α. The same section reports numerical observations of non-monotonicity for large α. Theorems 3.4 and 4.1 give the λ→∞ asymptotics and bounds: Shannon, Rényi, and both generalized Rényi entropies diverge logarithmically; Tsallis entropy grows like a power for α∈(0,1) and converges to 1/(α−1) for α>1; Sharma–Mittal entropy does the same with β in place of α.

Load-bearing premise

Proposition 5.4 assumes that a positive continuously differentiable function on (0,1) which tends to 0 at 0 and equals 0 at 1 must be increasing on some interval near 0; continuity and endpoint limits alone do not force this, and the paper supplies no further analytic information about ρ(α).

Editorial extensions

If this is right

  • For a Poisson count model, the Tsallis and Sharma–Mittal entropies can be used as uncertainty measures that are guaranteed to grow with the rate λ, just as Shannon and Rényi entropies do.
  • The one-parameter generalized Rényi entropy is not a universally reliable uncertainty measure: for α in a small interval near 0, increasing λ can lower the entropy over an interval starting at λ=1.
  • For the two-parameter generalized Rényi entropy, any parameter pair close to an anomalous α inherits the decrease, so the phenomenon is not isolated to a single parameter choice.
  • The asymptotic formulas give practical approximations: for large λ, Shannon, Rényi, and generalized Rényi entropies grow like (1/2)log(2πλ) plus parameter-dependent constants, while Tsallis (0<α<1) and Sharma–Mittal (0<β<1) grow as power laws.
  • Two-sided estimates, e.g. for Shannon entropy L(λ) ≤ H_SH(λ) ≤ U_SH(λ), control the entropies uniformly for λ>1 and agree with the leading asymptotic term.

Reading between the lines

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

  • The derivative identities used here—expressing each entropy through ψ and log ψ—are general; for any discrete distribution where ψ(α,λ) can be controlled, the same argument would classify monotonicity, so the Poisson results are likely a template rather than an isolated example.
  • The large-α anomaly is presented numerically only; proving it analytically, for example by showing that the oscillating derivative ∂_λ H_GR(α,λ) crosses zero for every sufficiently large α, remains an open extension the paper does not attempt.
  • In Poisson applications, the decreasing interval near λ=1 for small α means that a generalized Rényi-based score can fall when more events are observed, so practitioners should check monotonicity for their parameter range before using H_GR as an uncertainty measure.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

1 major / 6 minor

Summary. The paper studies six entropy functionals—Shannon, Rényi, Tsallis, Sharma–Mittal, and two generalized Rényi entropies—for the Poisson distribution with intensity λ. It expresses all of them through the function ψ(α,λ)=Σ_i p_i(λ)^α, derives explicit formulas, asymptotic expansions as λ→∞, upper and lower bounds, and monotonicity results in λ. The main claims are that Shannon, Rényi, Tsallis, and Sharma–Mittal entropies are strictly increasing in λ for all admissible parameters, while the generalized Rényi entropies can be non-monotone for α near 0 (proved) and for large α (presented as numerical evidence), with the two-parameter version inheriting this non-monotonicity locally.

Significance. If correct, the monotonicity and anomaly results constitute a rigorous, parameter-free characterization of entropy behavior for Poisson count data, with practical cautionary value for researchers using generalized Rényi entropies as uncertainty measures. The ψ-based proof strategy is clean and yields explicit asymptotic constants; the main monotonicity theorems are correct, the α-near-0 anomaly proof is valid although terse, and the large-α anomaly is appropriately labeled numerical. The paper also provides useful explicit bounds, but the bound theorem contains a false statement for the Tsallis and Sharma–Mittal entropies in the α,β>1 regime; this does not undermine the monotonicity theorems but must be corrected before acceptance.

major comments (1)
  1. [Theorem 4.1, items 4–5 (proof on p. 10; eqs. (2.13), (3.14)–(3.15), (4.6))] For α>1 and β>1, the stated replacements are false. Combining (4.6) with (2.13) gives H_T(α,λ) ≥ 1/(α−1)[1−(2πλ)^{−(α−1)/2} e^{(α−1)h(λ)}], not the factor e^{−h(λ)} claimed in item 4; the analogous factor for H_SM(α,β,λ) in item 5 is e^{(β−1)h(λ)}, not e^{−h(λ)}. The stated stronger bound is numerically violated: at α=2, λ=1, H_T(2,1)≈0.6915, while the claimed lower bound 1−(2π)^{−1/2}e^{−h(1)}≈0.6954. For 0<α<1 and 0<β<1 the claimed bounds are true but weaker than what follows from (4.6); the theorem should state the correct exponential factors for each parameter range.
minor comments (6)
  1. [Proposition 5.4, proof] The inference that ρ must be increasing on an interval near α=0 is terse; please spell out the argument: for δ>0, the mean value theorem gives c∈(0,δ) with ρ′(c)=ρ(δ)/δ>0, and continuity of ρ′ then yields an interval around c, which can be chosen inside an arbitrarily small neighborhood of 0.
  2. [Proposition 2.3, proof step 3] In the proof of the lower bound for H_GR(α,β,p), the displayed expression should be −log μ(p), not μ(p).
  3. [Remark 5.2] For α>1 the text says the entropy is bounded by/increases to 1/(1−α); this should read 1/(α−1).
  4. [Abstract and Remark 5.7] The anomalous behavior for large α is numerical evidence, not a theorem; please state explicitly in the abstract or introduction that the proved anomaly covers α near 0, while the large-α anomaly is observed numerically.
  5. [Lemma A.6, part 1] The initial restriction γ∈(α,1) is incompatible with the later uniform choice γ≥γ* when α>γ*; rephrase the proof to allow γ∈(0,1) and rely on the uniform argument at the end of the lemma.
  6. [Proposition 5.8, proof of item 1] The proof covers β>1>α and β>1=α; please state explicitly that the α>1≥β case follows by the symmetry H_GR(α,β,λ)=H_GR(β,α,λ).

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the monotonicity and anomaly results are derived from published, parameter-free properties of ψ(α,λ), and the anomalous behavior is not fitted or defined into existence.

full rationale

The paper's central claims are not circular. The Tsallis and Sharma–Mittal monotonicity results follow algebraically from the monotonicity of ψ(α,λ) in λ, a result quoted from the authors' prior work [9, Theorem 2]. That prior result is published, parameter-free, has stated assumptions that do not include the target monotonicity of Tsallis or Sharma–Mittal entropies, and is not equivalent to the present claims, so the self-citation is legitimate independent support rather than circularity. The anomalous behavior of H_GR(α,λ) is derived from an explicit derivative identity, ∂_λ H_GR = −∂_α∂_λ log ψ, and the proof of Proposition 5.4, while terse, is mathematically valid: strict positivity of ρ(δ) together with ρ(0)=0 gives, via the mean value theorem, a point c with ρ′(c)>0, and continuity gives an interval on which ρ′>0; no fitted parameter or target quantity is smuggled in. The large-α anomalous behavior in Remark 5.7 is explicitly labeled numerical and is not presented as a theorem, so it cannot constitute a circular derivation. There is no self-definitional step, no fitted input renamed as a prediction, no uniqueness theorem imported from the authors, and no ansatz hidden in a citation. The apparent error in Theorem 4.1 items 4–5, where substituting (4.6) into (2.13) yields a different exponent than claimed, is a correctness issue, not a circularity issue, and does not affect the monotonicity or anomaly derivations.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The paper introduces no new physical entities and fits no parameters to data. Its main analytic tools are standard Stirling bounds, term-by-term differentiation of ψ, and saddle-point asymptotics. The most fragile axiom is the unproved monotonicity of ρ near α=0, which underlies the anomalous small-α claim.

assumptions (4)
  • standard math Stirling-type bounds for n! and Γ(x+1), used throughout Sections 3 and 4
    Invoked in Lemma A.2, Lemma A.6, and Lemma A.7 to control Poisson probabilities, gamma ratios, and asymptotic growth.
  • standard math Term-by-term differentiation and uniform convergence of the series defining ψ(α,λ)
    Lemma A.4 justifies the derivative formulas (3.7), (3.8), and (5.1), which are central to the monotonicity analysis.
  • domain assumption Saddle-point approximation replacing the sum in ψ(α,λ) by an integral with controlled remainder
    Lemma A.7 derives the central asymptotic ψ(α,λ)∼(1/√α)(2πλ)^{(1-α)/2} using the saddle-point method, but explicit error bounds are not given.
  • ad hoc to paper The function ρ(α) is increasing on some interval near α=0
    Proposition 5.4 asserts this from positivity and endpoint limits of ρ. This is not a theorem for general C^1 functions, and the paper does not prove analyticity or provide a derivative expansion for ρ.

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Pith. "Pith review of Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior." pith.science (2026). https://pith.science/paper/HIVGFUFE

@misc{pith2026241116913,
  author       = {Pith},
  title        = {Pith review of: Entropies of the Poisson distribution as functions of intensity: "normal" and "anomalous" behavior},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HIVGFUFE}},
  note         = {Machine review of arXiv:2411.16913}
}
abstract

The paper extends the analysis of the entropies of the Poisson distribution with parameter $\lambda$. It demonstrates that the Tsallis and Sharma-Mittal entropies exhibit monotonic behavior with respect to $\lambda$, whereas two generalized forms of the R\'enyi entropy may exhibit "anomalous" (non-monotonic) behavior. Additionally, we examine the asymptotic behavior of the entropies as $\lambda \to \infty$ and provide both lower and upper bounds for them.

Figures

Figures reproduced from arXiv: 2411.16913 by the authors.

Figure 1
Figure 1. A comparison of the Shannon entropy HSH(λ) with the lower estimate L(λ) given by (4.1), the upper estimate USH(λ) given by (4.3), and the asymptotic function ASH(λ) defined by the right hand side of (3.11). Remark 4.2. We are going to illustrate the obtained estimates. 1. The lower and upper bounds (4.1) and (4.3) for the Shannon entropy are agreed with the leading term 1 2 log(2πλ) of the asymptotics (3.11). Surpri… view at source ↗
Figure 2
Figure 2. A comparison of the R´enyi entropy HR(α, λ) with the lower estimate L(λ) given by (4.1), the upper estimate UR(α, λ, γ) given by (4.4)–(4.5) (with optimal values of γ for λ ∈ [1, 20]), and the asymptotic function AR(α, λ) defined by the right hand side of (3.12). 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. The Tsallis entropy HT (α, λ) (a) HSM(2, β, λ) as a function of β and λ 5 10 15 20 λ 2 4 6 8 HSM β = 0.3 β = 0.6 β = 0.9 β = 1.2 β = 2 β = 3 (b) HSM(2, β, λ) as a function λ for various β [PITH_FULL_IMAGE:figures/full_fig_p014_3.png] view at source ↗
Figures from the paper (10 more)
Figure 4
Figure 4. Figure 4: The Sharma–Mittal entropy for α = 2 (a) HSM(α, 2, λ) as a function of α and λ 5 10 15 20 λ 0.2 0.4 0.6 0.8 1.0 HSM α = 0.3 α = 0.6 α = 0.9 α = 1.2 α = 2 α = 3 (b) HSM(α, 2, λ) as a function λ for various α [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: The Sharma–Mittal entropy for β = 2 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: The Sharma–Mittal entropy for β = 0.5 5.2 “Anomalous“ behavior of generalized R´enyi entropies We now turn to the study of two generalized R´enyi entropies, which exhibit “anomalous” be￾havior for certain values of the parameters. 5.2.1 The generalized R´enyi entropy H…
Figure 7
Figure 7. Figure 7: Graphs of the function ρ(α) and its derivative ρ ′ (α). Remark 5.6 [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: The generalized R´enyi entropy HGR(α, λ): (a) as a function of (α, λ), (b) as a function of λ for α = 0.1, (b) as a function of λ for α = 0.14 Remark 5.7. We can also observe numerically that the generalized R´enyi entropy HGR(α, λ) demonstrates “anomalous” behavior fo…
Figure 9
Figure 9. Figure 9: The non-monotone behavior of HGR(α, λ) in λ for large α 5.2.2 The generalized R´enyi entropy HGR(α, β, λ) Now we consider the generalized R´enyi entropy with two parameters α and β. Proposition 5.8. 1. Let α ≤ 1 < β or α > 1 ≥ β. Then HGR(α, β, λ) increases as a functi…
Figure 10
Figure 10. Figure 10: Damping oscillations of ∂ ∂λHGR(α, λ) in λ for different values of α (dependent on α). Then, for each α ∈ J and for each β in a neighborhood of α, the entropy HGR(α, β, λ) is also decreasing in λ on some interval (dependent on α, β). Proof. First, we note that, by (3.…
Figure 8
Figure 8. Figure 8: The proof of the second statement of Proposition 5.8 implies that the behavior of HGR(α, β, λ) for large α and β should be similar to the one observed in [PITH_FULL_IMAGE:figures/full_fig_p020_8.png]
Figure 11
Figure 11. Figure 11: The generalized R´enyi entropy HGR(α, β, λ): (a) as a function of (α, λ) for β = 0.01, (b) as a function of λ for α = 0.02 and β = 0.01 A Appendix A.1 Auxiliary results Lemma A.1. For any λ > 0, lim λ→∞ e −λX∞ i=1 λ i i! log(i + 1) = ∞. Proof. Obviously, for any N > 1…
Figure 12
Figure 12. Figure 12: The generalized R´enyi entropy HGR(α, β, λ) for large α and β Furthermore, lim λ→∞ e −λ N X−1 i=0 λ i i! = 0. Therefore, lim λ→∞ e −λX∞ i=1 λ i i! log(i+1) ≥ log(N +1) lim λ→∞ e −λ X∞ i=N λ i i! = log(N +1) lim λ→∞ e −λX∞ i=0 λ i i! = log(N +1). Since N > 1 is arbitra…

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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.