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REVIEW 4 major objections 5 minor 28 references

Diagrammatics of information

T0 review · 4 major / 5 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read Shannon entropy can be encoded in diagrams whose algebraic symbols satisfy the same 4-term relation as the infinitesimal dilogarithm, and the 5-term dilogarithm collapses to that relation under a dual-number substitution.

desk verdict The entropy part is a clean, if elementary, exposition; the two advertised complete proofs of the dilogarithm deformation are not complete, and the second one contains a concrete algebraic error. read the letter →

arxiv 2502.01983 v1 pith:5FDF6CMZ submitted 2025-02-04 math-ph cs.ITmath.ITmath.MP

Mee Seong Im , Clement Kam , Caden Pici This is my paper · ORCID
classification math-phcs.ITmath.ITmath.MP MSC 57K1618M3028D2037A3568P3094A1594A40
keywords Shannonentropyjointconditionalmutualinformationdiagrammaticalgebradilogarithminfinitesimaldualnumbers
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 claims that Shannon entropy, joint entropy, and conditional entropy can be read off from planar networks of additive and multiplicative lines: each network evaluates to a sum of formal symbols $\langle a,b\rangle$ that obey a 2-cocycle relation, and when all lines of a probability distribution merge, that sum is exactly $H(p_X)=-\sum_i p_i\log p_i$. This connects information theory to a vector-space structure whose relations are the same as those of the infinitesimal dilogarithm. The paper's main new result is a written-out degeneration: substituting $\langle a\rangle = a+a(1-a)t$ into the 5-term dilogarithm relation over the dual numbers $k[t]/(t^2)$ and taking $t\to 0$ yields the 4-term infinitesimal dilogarithm relation $[a]-[b]+a[b/a]+(1-a)[(1-b)/(1-a)]=0$, and the paper claims two complete proofs of this degeneration. A sympathetic reading takes these proofs as establishing a direct algebraic bridge from classical entropy to dilogarithm identities.

What carries the argument

The load-bearing object is the pair of vector spaces $J(k)$ and $\beta(k)$. $J(k)$ is spanned by formal symbols $\langle a,b\rangle$ with the relations $\langle a,b\rangle=\langle b,a\rangle$, $\langle ca,cb\rangle=c\langle a,b\rangle$, and $\langle a,b+c\rangle+\langle b,c\rangle=\langle a+b,c\rangle+\langle a,b\rangle$; $\beta(k)$ is spanned by symbols $[a]$ subject to the 4-term relation above. The diagrammatic calculus gives a third presentation: additive black lines merge and split with evaluations $\pm\langle a,b\rangle$, multiplicative red lines rescale labels, and the value of a diagram is the sum over its additive vertices. The deformation is carried by the ring of dual numbers $k[t]/(t^2)$: every $a\in k^*\setminus\{1\}$ is sent to the invertible element $\langle a\rangle=a+a(1-a)t$, and lemmas comparing quotients such as $(1-\langle b\rangle)/(1-\langle a\rangle)$ convert the 5-term relation into the 4-term relation as $t\to 0$.

What would settle it

Reread the first proof of Proposition 10.1 and try to locate the map $D\colon \beta(k)\to k^{*}\otimes k^{+}$; it is used in a commutative diagram but never defined, and neither $D$ nor $\rho$ is shown injective. Finding a nonzero element of $\beta(k)$ that $D$ or $\rho$ kills, or confirming that the cited $[Zagier]$ reference has no bibliography entry, would show that the paper's claim of two complete proofs is not yet complete.

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Extended reading notes

Core claim

The central claim is that entropy and the infinitesimal dilogarithm live in the same algebraic structure. In the diagrammatic calculus, a probability distribution is a collection of additive lines that merge into one line; summing the contributions of the merging vertices, weighted by multiplicative lines that rescale labels, gives Shannon entropy. The same symbols $\langle a,b\rangle$ satisfy the relations of the vector space $J(k)$, and the 4-term relation $[a]-[b]+a[b/a]+(1-a)[(1-b)/(1-a)]=0$ is the entropy functional equation written in these symbols. The paper's new technical result is the deformation: in the ring $k[t]/(t^2)$, the substitution $\langle a\rangle=a+a(1-a)t$ turns the 5-term dilogarithm relation into an equation whose $t\to 0$ limit is exactly the 4-term infinitesimal dilogarithm relation, and the paper asserts this is proved in two independent ways.

Load-bearing premise

The first proof of the deformation assumes that two auxiliary maps are injective, yet one of those maps is never actually defined; if that assumption fails, the first proof does not establish the isomorphism between the infinitesimal dilogarithm space and the deformed space.

Editorial extensions

If this is right

  • Every diagram with fixed boundary labels evaluates to the same entropy value, so information quantities can be studied as boundary invariants of planar networks.
  • Joint entropy and conditional entropy appear by grouping lines in the same network, which recovers the chain rule $H(X,Y)=H(X)+H(Y\mid X)$ by diagrammatic means.
  • The degeneration result pins down the 4-term infinitesimal dilogarithm as the first-order coefficient of the 5-term dilogarithm under the dual-number substitution, making the classical-to-infinitesimal passage explicit.
  • Because $J(k)$ and $\beta(k)$ are isomorphic, entropy, joint entropy, and the 2-cocycle structure all fit into one vector space, giving a single algebraic setting for information-theoretic identities.

Reading between the lines

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

  • A natural next test is to draw explicit diagrams for mutual information and verify that the 2-cocycle relation reproduces $I(X;Y)=H(X)+H(Y)-H(X,Y)$; the paper leaves this as an exercise.
  • The same dual-number trick could be applied to higher polylogarithm relations, producing an infinitesimal family of identities indexed by the order of the polylogarithm; the paper does not pursue this.
  • If the missing injectivity argument for the map $D$ in the first proof is supplied, the isomorphism $\beta(k)\cong TP(k)$ would give a direct interpretation of entropy as a first-order cocycle; that consequence is implicit in the paper's structure but not stated.
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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

4 major / 5 minor

Summary. The paper presents a diagrammatic treatment of Shannon entropy following the first author's earlier work with Khovanov, and connects these diagrams to Cathelineau's vector space J(k), Kontsevich's entropy functional equation, joint entropy, and conditional entropy. The stated novel contribution is two complete proofs that the 5-term dilogarithm relation, evaluated on dual numbers, deforms to the 4-term infinitesimal dilogarithm relation in the limit t→0. Sections 2–9 recall and reinterpret existing results, with explicit proofs of several functional identities (Propositions 3.2, 3.6, 4.2) and diagrammatic evaluations. Section 10 attempts two proofs of the deformation claim.

Significance. If the deformation claim were fully established with complete, rigorous proofs, the paper would provide a useful expository bridge between diagrammatics for entropy and infinitesimal dilogarithms, complementing Cathelineau's and Kontsevich's work. The entropy identities in Sections 3–4 are standard but are here verified in detail, and the diagrammatic language (mostly from [IK24a]) is presented clearly for an information-theory audience. However, the advertised 'two complete proofs' are not complete: both arguments in Section 10 contain load-bearing gaps. Since the paper's abstract and title emphasize these proofs as a main contribution, these gaps materially affect the paper's value.

major comments (4)
  1. [§10, first proof of Proposition 10.1] The proof of injectivity of φ relies on a commutative diagram in which a map D: β(k) → k* ⊗ k+ appears, but D is never defined. The text states that D and ρ are injective, yet no proof of either injectivity claim is given, and ρ is only defined by the formula in (10.4). Without a definition of D and a proof of injectivity of both maps, the claimed isomorphism β(k) ≅ TP(k) is not established. This is a load-bearing gap in the first proof.
  2. [§10, second proof of Proposition 10.9] The second proof does not correctly perform the t→0 degeneration. After applying Lemmas 10.3–10.7, equation (10.5) becomes [⟨a⟩] − [⟨b⟩] + a[⟨b/a⟩] + [(1−b^{-1})/(1−a^{-1})] + (1−a)[⟨(1−b)/(1−a)⟩] = 0. The proof then asserts that the element [(1−b^{-1})/(1−a^{-1})] is the zero element in ker(β(k2)→β(k)) modulo certain relations. But that element is not in the kernel: its image under reduction t→0 is the nonzero class [(1−b^{-1})/(1−a^{-1})] in β(k). Consequently, applying the reduction to the whole displayed equation yields the 5-term relation in β(k), not the 4-term relation (10.6). To obtain (10.6) one must divide by the t-linear part and kill all constant terms while preserving [b/a] and [(1−b)/(1−a)], but no such quotient is defined. The sentence 'which t-deforms to' is therefore not justified, and the second proof fails independently of the first.
  3. [§4, Theorem 4.3] Theorem 4.3, the general formula for joint entropy H(p_X,p_Y) as a sum of ⟨·,·⟩ symbols, is asserted without proof: 'The proof of Theorem 4.3 is a direct calculation, similar to the proof of Proposition 3.6 and Proposition 4.2.' Since this theorem is one of the paper's substantive new formulas (the two-variable case is only Proposition 4.2), a complete proof or a precise reference is needed. As written, the result is unsupported.
  4. [§7, Proposition 7.1] The key invariance property Φ(γ) depends only on the source and target objects is cited from [IK24a] with no proof or even a sketch. This property is foundational for the diagrammatic calculus used throughout Sections 5–8, and the paper would be more self-contained if the proof or at least the mechanism of the invariance were included.
minor comments (5)
  1. [Title of §3] The section title reads 'Shannon entropy and and Cathelineau’s vector space'; the duplicated 'and' should be removed.
  2. [§2.1, Lemma 2.1] In the proof of Lemma 2.1, the equality ⟨a,−a⟩ = a⟨1,−1⟩ uses the scaling relation; this is fine, but the line 'So ⟨1,−1⟩ = ⟨−(−1),−1⟩ = −⟨−1,1⟩' would be clearer if the scaling relation were explicitly cited at each step.
  3. [§3.1, Figure 3.1.1] The figure caption states the graph is for −1 ≤ p ≤ 2, but the entropy function is only defined via (3.3) for all real p; consider noting that the extension is by the absolute-value formula shown.
  4. [§4.1, Proposition 4.1] The proof of Proposition 4.1 is omitted with the phrase 'similar to the proof of Lemma 3.3.' Given that the indexing p_{ij} is new, a brief verification of the 2-cocycle relation would help.
  5. [§10, Lemma 10.7] Lemma 10.7 states scaling identities [b ∗ ⟨c⟩] = b[⟨c⟩] and related identities are 'clear using the construction of β(k).' These identities involve the action of k on the dual numbers and are essential to the proof; a few lines of justification would remove any ambiguity.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the deformation claims rest on explicit algebraic identities; identified gaps are correctness issues, not definitional circularity.

full rationale

The paper's derivation chain does not reduce to its own inputs by construction. The 5-term-to-4-term deformation is attempted through explicit dual-number substitutions, and the target 4-term relation is not assumed as an input of that derivation. The first proof of Proposition 10.1 has a serious omitted-proof issue: the map D is never defined and the injectivity of D and ρ is asserted without proof, so the claimed isomorphism β(k) ≅ TP(k) is not established. The second proof of Proposition 10.9 drops the summand [(1-b^{-1})/(1-a^{-1})] as a zero element in a kernel, but its image under evaluation at t = 0 is the generally nonzero class [(1-b^{-1})/(1-a^{-1})] in β(k); this makes the t → 0 step invalid as written. However, these are gaps or errors in a proof attempt, not cases where a prediction is equivalent to an input by construction, nor where a fitted parameter is renamed as a prediction. The diagrammatic framework is explicitly credited to prior independent work [IK24a], and Proposition 7.1 is cited rather than derived from the paper's conclusions; self-citation of that kind is not load-bearing circularity here. The entropy identities in Sections 3 and 4 are direct calculations from the definition of H, not circular or fitted. Accordingly, the appropriate finding is no significant circularity, with the noted mathematical gaps weighed as correctness risks rather than circularity.

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

The central claims rest on standard background from Cathelineau and Im-Khovanov, plus several unproved or under-proved injectivity and scaling facts that are specific to this paper's proofs. The entropy functional relations are standard. There are no fitted parameters or invented physical entities.

assumptions (6)
  • domain assumption The field k in Section 2 is assumed to be of characteristic 0.
    The whole construction of J(k) and beta(k) and the isomorphism H(k) to J(k) (Theorem 3.5) uses characteristic 0 to conclude that the symbols vanish appropriately.
  • domain assumption The entropy function H(p) = -p log|p| - (1-p) log|1-p| satisfies the 4-term functional equation (3.4) and is continuous on R.
    Stated as known from Faddeev and Kontsevich; proved in Proposition 3.2 but the proof uses standard logarithm identities.
  • domain assumption The diagrammatic invariant Phi(gamma) depends only on the source and target objects (Proposition 7.1).
    Cited from [IK24a, Section 5] without proof; all diagrammatic entropy evaluations in Sections 5, 6 and 8 assume this invariance.
  • ad hoc to paper The map D: beta(k) to k* tensor k+ in the first proof of Proposition 10.1 exists and is injective.
    D is not defined and its injectivity is asserted, not proved or referenced; this is load-bearing for proving phi injective.
  • ad hoc to paper The map rho: TP(k) to k* tensor k+ in (10.4) is injective.
    Asserted without proof in the first proof of Proposition 10.1; without it, phi is not shown injective.
  • ad hoc to paper Lemma 10.7: scaling identities [b * <c>] = b[<c>], (-1)[<1-a>] = -[<a>], and a[<a^{-1}>] = -[<a>] hold in beta_2(k_2).
    The proof says 'This is clear using the construction of beta(k)', but no detailed verification is given, and these identities are used in the final deformation step.

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Pith. "Pith review of Diagrammatics of information." pith.science (2026). https://pith.science/paper/5FDF6CMZ

@misc{pith2026250201983,
  author       = {Pith},
  title        = {Pith review of: Diagrammatics of information},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5FDF6CMZ}},
  note         = {Machine review of arXiv:2502.01983}
}
abstract

We introduce a diagrammatic perspective for Shannon entropy created by the first author and Mikhail Khovanov and connect it to information theory and mutual information. We also give two complete proofs that the $5$-term dilogarithm deforms to the $4$-term infinitesimal dilogarithm.

Figures

Figures reproduced from arXiv: 2502.01983 by the authors.

Figure 3.1
Figure 3.1. 1. Entropy function H(p) = −p log ∣p∣ − (1 − p) log ∣1 − p∣ for −1 ≤ p ≤ 2. 3. Shannon entropy and and Cathelineau’s vector space 3.1. Shannon entropy. Let X = {x1, . . . , xn}, a finite set. Shannon entropy of a finite probability distribution pX on X that associates probabilities p1, . . . , pn, where n ∑ i=1 pi = 1, 0 < pi < 1 to its points xi , respectively, is given by (3.1) H(pX) = − n ∑ i=1 pi log pi . We can… view at source ↗
Figure 5.0
Figure 5.0. 1. Upper left: Whenever two black additive lines merge, we eval￾uate ⟨a, b⟩ at the additive vertex. Upper middle: whenever two additive lines split, we evaluate −⟨a, b⟩. Upper right: we are allowed to have virtual cross￾ing whenever two additive lines cross but the intersection of these two lines is virtual, so there is no corresponding evaluation for these two additive lines. Bottom left: whenever a red line is to … view at source ↗
Figure 5.0
Figure 5.0. 2. When additive lines are pointing downwards, we rotate the additive vertices in the network whilst fixing the boundary points so that the orientations at the additive vertices are upwards. Top row: the additive vertex in each of the figures gives the contribution of ⟨a, b⟩. Bottom row: the additive vertices give −⟨a, b⟩. Proof. Recall from [IK24a] that a relation between diagrammatics and scaled entropy is (3.6). … view at source ↗
Figures from the paper (9 more)
Figure 5.0
Figure 5.0. Figure 5.0: 3. Top left: This diagram represents the 2-cocycle condition (3) in Cathelineau’s k-vector space. Top right: Two lines crossing is virtual, but the additive vertex contributes ⟨b, a⟩ = ⟨a, b⟩. Bottom left: the additive vertex on the left hand side gives ⟨ac, bc⟩ whil…
Figure 5.0
Figure 5.0. Figure 5.0: 5. All possible ways of network of additive lines merging into one additive line. ⋯ p1 p2 . . . pn p1 + p2 + . . . + pn = ⋯ p1 p2 . . . pn p1 + p2 + . . . + pn = ⋯ p1 p2 . . . pn p1 + p2 + . . . + pn [PITH_FULL_IMAGE:figures/full_fig_p018_5_0.png]
Figure 5.0
Figure 5.0. Figure 5.0: 8. We introduce a dot to represent reversal of orientation. Left: we can erase the 0-line. The 0-line can go in or out, to the top or bottom boundary. Right: we have ⟨p,−p⟩ = 0. p −p = p −p [PITH_FULL_IMAGE:figures/full_fig_p019_5_0.png]
Figure 6.0
Figure 6.0. Figure 6.0: 1. Left: the blue boundary wall absorbs the additive defect on the a-line and b-line, emitting a floating point labeled (a + b) [ a a + b ]. Right: The three additive vertices are absorbed, emitting one vertex with the label [b]. b a − b b − a 1 − b a 1 − a 1 = a [ b…
Figure 6.0
Figure 6.0. Figure 6.0: 3. If p1 + p2 + . . . + pn = 1, the blue boundary wall absorbs all additive vertices, giving us entropy. larger blue dotted ellipse, we have the theory of probability. When networks turn around, pointing downwards, and when we can rescale the additive lines, we have …
Figure 6.0
Figure 6.0. Figure 6.0: 4. The diagram shows a morphism (cobordism). Small purple ellipse encloses a part of the morphism that admits an interpretation in ear￾lier categorical approaches to entropy. Larger blue ellipse encloses a part of the diagram that can be interpreted in classical prob…
Figure 8.0
Figure 8.0. Figure 8.0: 1. Joint entropy for two random variables X and Y with proba￾bilities p1 and p2 and q1 and q2, respectively. p1 p2 p3 q2 q1 y2 ∶= p12 + p22 + p32 p12 + p22 p11 + p21 y1 ∶= p11 + p21 + p31 p12 p22 p32 p11 p21 p31 [PITH_FULL_IMAGE:figures/full_fig_p023_8_0.png]
Figure 8.0
Figure 8.0. Figure 8.0: 3. Let {p1, p2, p3} be probabilities for the random variable X and {q1, q2, q3} be probabilities for the random variable Y . Let pij ∶= (pi , qj). Top left: random variable X with probabilities {p1, p2, . . . , pk}. Top right: random variable Y with probabilities {q1…
Figure 8.0
Figure 8.0. Figure 8.0: 4. Let {p1, p2, p3} be probabilities for the random variable X and {q1, q2, q3} be probabilities for the random variable Y . Let pij ∶= (pi , qj). Above is a diagrammatic of 0 since the top and the bottom collections of boundary points in it are identical. We leave i…

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Works this paper leans on

28 extracted references · 27 canonical work pages

  1. [1]

    Leon Brillouin, Science and information theory, Academic Press, Inc., New York, 1956

    work page 1956

  2. [2]

    Jean-Louis Cathelineau, Sur l'homologie de SL_2 \`a coefficients dans l'action adjointe , Math. Scand. 63 (1988), no. 1, 51--86

    work page 1988

  3. [3]

    Algebra 332 (2011), 87--113

    , Infinitesimal dilogarithms, extensions and cohomology, J. Algebra 332 (2011), 87--113

    work page 2011

  4. [4]

    David Deutsch and Patrick Hayden, Information flow in entangled quantum systems, R. Soc. Lond. Proc. Ser. A Math. Phys. Eng. Sci. 456 (2000), no. 1999, 1759--1774

    work page 2000

  5. [5]

    Faddeev, On the concept of entropy of a finite probabilistic scheme, Uspehi Mat

    Dmitry K. Faddeev, On the concept of entropy of a finite probabilistic scheme, Uspehi Mat. Nauk (N.S.) 67 (1956), no. 1, 227--231

    work page 1956

  6. [6]

    Paul Gustafson, Mee Seong Im, Remy Kaldawy, Mikhail Khovanov, and Zachary Lihn, Automata and one-dimensional TQFT s with defects , Lett. Math. Phys. 113 (2023), no. 93, 1--38

    work page 2023

  7. [7]

    Paul Gustafson, Mee Seong Im, and Mikhail Khovanov, Boolean TQFT s with accumulating defects, sofic systems, and automata for infinite words , Lett. Math. Phys. 114 (2024), no. 6, Paper No. 135

    work page 2024

  8. [8]

    Foams with flat connections and algebraic K-theory

    David Gepner, Mee Seong Im, Mikhail Khovanov, and Nitu Kitchloo, Foams with flat connections and algebraic k-theory, arXiv preprint arXiv:2405.14465, submitted to Ann. K-Theory (2024), 1--57

    work page Pith review arXiv 2024

Show all 28 references
  1. [9]

    Hin cin, On the basic theorems of information theory, Uspehi Mat

    Aleksandr Y. Hin cin, On the basic theorems of information theory, Uspehi Mat. Nauk (N.S.) 11 (1956), no. 1(67), 17--75

    1956

  2. [10]

    Mee Seong Im and Mikhail Khovanov, Foams, iterated wreath products, field extensions and S ylvester sums , arXiv preprint arXiv:2107.07845 https://arxiv.org/abs/2107.07845 (2021), 1--78

    2021 arXiv

  3. [11]

    , Topological theories and automata, arXiv preprint arXiv:2202.13398 https://arxiv.org/abs/2202.13398 (2022), 1--70

    2022 arXiv

  4. [12]

    , From finite state automata to tangle cobordisms: a TQFT journey from one to four dimensions , arXiv preprint arXiv:2309.00708 https://arxiv.org/abs/2309.00708, to appear in Contemp. Math. (2023), 1--40

    2023 arXiv

  5. [13]

    , Entropy, cocycles, and their diagrammatics, arXiv preprint arXiv:2409.08462 https://arxiv.org/abs/2409.08462 (2024), 1--82

    2024 arXiv

  6. [14]

    , Foam cobordism and the S ah- A rnoux- F athi invariant , arXiv preprint arXiv:2403.06030 https://arxiv.org/abs/2403.06030 (2024), 1--38

    2024 arXiv

  7. [15]

    , One-dimensional topological theories with defects: the linear case, Contemporary Mathematics 791 (2024), 105--146

    2024

  8. [16]

    Mee Seong Im, Mikhail Khovanov, and Victor Ostrik, Universal construction in monoidal and non-monoidal settings, the B rauer envelope, and pseudocharacters , Theory Appl. Categ. 44 (2025), no. 2, 15--83

    2025

  9. [17]

    2, 319--331

    Mee Seong Im and Paul Zimmer, One-dimensional topological theories with defects and linear generating functions, Involve 15 (2022), no. 2, 319--331

    2022

  10. [18]

    Khinchin, Mathematical foundations of information theory, Dover Publications, Inc., New York, 1957, Translated by R

    Aleksandr I. Khinchin, Mathematical foundations of information theory, Dover Publications, Inc., New York, 1957, Translated by R. A. Silverman and M. D. Friedman

    1957

  11. [19]

    Mikhail Khovanov, Universal construction of topological theories in two dimensions, arXiv preprint arXiv:2007.03361 https://arxiv.org/abs/2007.03361 (2020), 1--56

    2020 arXiv

  12. [20]

    A ppendix to: `` O n poly(ana)logs

    Maxim Kontsevich, The 1 12 -logarithm. A ppendix to: `` O n poly(ana)logs. I '' [ C ompositio M ath 130 (2002), no. 2, 161--210] by P . E lbaz- V incent and H . G angl , Compositio Math. 130 (2002), no. 2, 211--214

    2002

  13. [21]

    Leff and Andrew F

    Harvey S. Leff and Andrew F. Rex, Maxwell's demon and the culture of entropy, Phys. Essays 10 (1997), no. 1, 125--149

    1997

  14. [22]

    Kumar Vijay Mishra, M. Ashok Kumar, and Ting-Kam Leonard Wong, Information G eometry for the W orking I nformation T heorist , arXiv preprint arXiv:2310.03884 https://arxiv.org/abs/2310.03884 (2023), 1--12

    2023 arXiv

  15. [23]

    12829, Springer, Cham, [2021] 2021

    Frank Nielsen and Fr\'ed\'eric Barbaresco (eds.), Geometric science of information, Lecture Notes in Computer Science, vol. 12829, Springer, Cham, [2021] 2021

    2021

  16. [24]

    P art I , Lecture Notes in Computer Science, vol

    Frank Nielsen and Fr\'ed\'eric Barbaresco (eds.), Geometric science of information. P art I , Lecture Notes in Computer Science, vol. 14071, Springer, Cham, [2023] 2023

    2023

  17. [25]

    P art II , Lecture Notes in Computer Science, vol

    Frank Nielsen and Fr\'ed\'eric Barbaresco (eds.), Geometric science of information. P art II , Lecture Notes in Computer Science, vol. 14072, Springer, Cham, [2023] 2023

    2023

  18. [26]

    Frank Nielsen, Geometric T heory of I nformation , Springer International Publishing Switzerland, Signals and Communication Technology, 2014

    2014

  19. [27]

    10, Paper No

    , An elementary introduction to information geometry, Entropy 22 (2020), no. 10, Paper No. 1100, 61

    2020

  20. [28]

    Inder Jeet Taneja, On the branching property of entropy, Ann. Polon. Math. 35 (1977/78), no. 1, 67--75

    1977

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