REVIEW 1 major objections 1 minor 1 cited by
Beyond integrated information: A taxonomy of information dynamics phenomena
T0 review · 1 major / 1 minor · reviewed 2026-08-14 · deepseek-v4-flash
Pith's one-line read A 16-atom decomposition of information flow shows why 'integration' is not one thing.
desk verdict A genuinely useful decomposition framework that resolves some cross-measure inconsistencies, but its taxonomy is a family of decompositions until the double-redundancy term is pinned down or shown irrelevant. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central object is the double-redundancy lattice $A \times A$, built as the product of the PID redundancy lattice for the source variables and for the target variables. Its 16 nodes $\alpha \to \beta$ carry values of a double-redundancy function $I^{\alpha \to \beta}_{\cap}$, and Möbius inversion over this lattice produces the 16 $\Phi$ID atoms $I^{\alpha \to \beta}_{\partial}$. Two axioms (compatibility with PID and Shannon terms, and monotonicity under the partial order) plus one free choice of the bottom atom $I^{\{1\}\{2\}\to\{1\}\{2\}}_{\partial}$ and a single-target redundancy function determine all atoms; the paper computes them numerically using a dependency-based unique-information measure.
What would settle it
Take the copy-transfer system defined in the paper (X1, X2, Y1 fair coin flips, Y2 = X1) and compute all 16 atoms with two different valid single-target redundancy functions, such as a minimum-redundancy function and the dependency-based $I_{\mathrm{dep}}$ measure. If either function assigns nonzero weight to any atom other than $I^{\{1\}\to\{2\}}_{\partial}$, the paper's claim that its example proofs hold for all $\Phi$IDs satisfying the axioms would fail, and the qualitative taxonomy would depend on an arbitrary choice.
Extended reading notes
Core claim
The central discovery is that the multivariate mutual information $E = I(X_1,X_2;Y_1,Y_2)$ of a two-variable Markovian process can be broken into 16 atoms arranged on a product of two PID redundancy lattices—one for the past and one for the future—via Möbius inversion of a double-redundancy function. These atoms correspond to qualitatively different dynamical phenomena: storage, copy, transfer, erasure, downward causation, and upward causation. The paper demonstrates that the widely used measure $\Phi_{\mathrm{WMS}}$ assigns the same value (1 bit) to three simple systems—copy transfer, downward XOR, and parity-preserving random—while their $\Phi$ID decompositions are completely different, with only $I^{\{1\}\to\{2\}}_{\partial}$, $I^{\{12\}\to\{1\}}_{\partial}$, or $I^{\{12\}\to\{12\}}_{\partial}$ nonzero respectively. It also shows that the four existing measures $\Phi_{\mathrm{WMS}}$, CD, $\psi$, and $\Phi_G$ respond to different subsets of atoms, implying they are not approximations of one concept but capture different phenomena.
Load-bearing premise
The axioms of $\Phi$ID do not determine the double-redundancy atom $I^{\{1\}\{2\}\to\{1\}\{2\}}_{\partial}$, and together with the choice of a single-target redundancy function this free choice fixes the numerical values of all 16 atoms.
Editorial extensions
If this is right
- Systems with identical $\Phi_{\mathrm{WMS}}$ can have qualitatively different information dynamics, so scalar integrated-information scores should not be read as measuring a single kind of integration.
- The taxonomy gives six disjoint phenomena—storage, copy, transfer, erasure, downward causation, and upward causation—that can be quantified separately from time-series data.
- Existing measures' inconsistencies are explained: each measure weights a different combination of atoms, so disagreements are in-principle, not just practical.
- The decomposition pinpoints why $\Phi_{\mathrm{WMS}}$ can be negative (a negative double-redundancy term) and suggests the corrected measure $\Phi_{\mathrm{WMS,c}}$.
- It diagnoses why unnormalised causal density can exceed total mutual information: it double-counts the downward-causation atom $I^{\{12\}\to\{1\}\{2\}}_{\partial}$; a system with $y_1=y_2=x_1\oplus x_2$ gives uCD = 2 bits vs. $E = 1$ bit.
Reading between the lines
- If the taxonomy is adopted, empirical studies could classify dynamical regimes by their atom profiles rather than by a scalar, for example comparing conscious versus unconscious brain states; the paper itself does not report such applications.
- Because the axioms leave the bottom atom unspecified, quantitative $\Phi$ID results may depend on the chosen redundancy function; comparing atom values across studies will require standardizing that choice or reporting robustness across choices.
- The framework is developed for bivariate Markovian systems, so extending the 16-atom picture to more variables or non-Markovian processes would require additional theory; the paper notes the Markovian restriction and leaves it for future work.
- The same decomposition could be used to construct new targeted integration measures for specific applications, choosing weights over atoms to match the phenomenon of interest; the paper suggests this possibility but does not build such measures.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper introduces Integrated Information Decomposition (ΦID), a framework that decomposes the multivariate mutual information E = I(X1,X2;Y1,Y2) of a bivariate Markovian system into 16 atoms organized on a product lattice of double-redundancies. The construction combines Williams and Beer's partial information decomposition (PID) with a second lattice over target variables, defining double-redundancy functions and using Möbius inversion to obtain atoms. The authors show that the resulting decomposition can be interpreted as a taxonomy of six information-dynamics phenomena (storage, copy, transfer, erasure, downward causation, upward causation), and they use it to argue that standard integrated-information measures such as ΦWMS conflate qualitatively different modes of information flow. They also provide example systems (copy transfer, downward XOR, parity-preserving random) with identical ΦWMS but different non-zero ΦID atoms, and they analyze why ΦWMS can be negative and why unnormalized causal density can exceed total mutual information.
Significance. If the construction is accepted, ΦID is a conceptually valuable extension of PID that offers a principled way to separate storage, transfer, and higher-order causal effects in multivariate time series. The paper's structural results—the lattice decomposition, the algebraic identities for ΦWMS and uCD, and the demonstration that different systems with the same scalar integrated information can differ in their information dynamics—are of genuine interest to the information dynamics and IIT communities. The proofs in the appendices are careful under the stated assumptions, and the authors are transparent about many limitations. The main open question is whether the qualitative taxonomy is invariant under the choice of redundancy function; this is the key issue that prevents the framework from being fully determinate.
major comments (1)
- [Appendices D and E] The proofs of the example systems and Table I rely on assumptions beyond Axioms 1 and 2: non-negativity of the double-redundancy function and the bound Red(X,Y;Z) ≤ min{I(X;Z), I(Y;Z)}. These are plausible and satisfied by common PID redundancy functions, but they are not stated as axioms of ΦID, and the paper does not show that they follow from Axioms 1–2. Consequently, the claims of Appendices D and E are conditional on a restricted class of ΦID instances, not on the framework as defined. The manuscript should specify that these results hold for any ΦID satisfying the stated extra conditions, and should make clear whether the authors regard these conditions as part of the definition of a 'valid' ΦID or as additional hypotheses for the examples.
minor comments (1)
- [Discussion] The claim that upward causation and synergistic storage 'have, to our knowledge, not been reported in the literature' is a strong novelty claim. The authors may wish to soften it or provide a more systematic literature search, since related ideas may exist in the synergy literature.
Circularity Check
No significant circularity: the ΦID construction is an explicit algebraic decomposition whose main qualitative claims are proved under stated axioms, not fitted to a target result.
full rationale
The paper's derivation chain is self-contained: Definition 1 and Eqs. (4)-(6) define the 16 ΦID atoms as the Möbius inversion of a double-redundancy function on the product lattice, and the two axioms fix the redundancies except one term. This is a mathematical construction rather than a disguised fit; no parameter is fitted to data and then relabelled as a prediction. The main claims about the three example systems in Fig. 3 are proved in Appendix D from only the partial-order axiom, non-negativity of the double-redundancy, and the standard PID bound Red ≤ min, so they do not depend on a particular redundancy function. Table I is obtained by decomposing the closed-form definitions of ΦWMS, CD, ψ, and ΦG into ΦID atoms, which is an algebraically checkable result rather than an empirical prediction forced by the framework. Proposition 1 and Appendix C openly acknowledge that the axioms do not uniquely determine all atoms and that a single-target PID redundancy function, plus one double-redundancy atom, must be supplied; this underdetermination is an explicit limitation and a source of convention-dependence, not a circular step. The only self-citation is Ref. [10] for the known inconsistent behaviour of integrated-information measures; it is used as background motivation and is independently reproducible, not as the load-bearing premise of the ΦID derivation. Hence no reduction of the central claim to its inputs was found.
Assumptions & free parameters
assumptions (5)
- ad hoc to paper Axiom 1 (compatibility): I^{α→β}_∩ reduces to PID redundancy or Shannon mutual information when one side has a single element.
- ad hoc to paper Axiom 2 (partial ordering): if α→β ≼ α'→β' then I^{α→β}_∩ ≤ I^{α'→β'}_∩.
- domain assumption The dynamics are Markovian, so E = I(X1,X2;Y1,Y2) for states at successive times.
- domain assumption A PID redundancy function Red(·) is chosen and is non-negative, with Red(X,Y;Z) ≤ min{I(X;Z), I(Y;Z)}.
- ad hoc to paper The existence of an unspecified double-redundancy function I^{{1}{2}→{1}{2}}_∂ that satisfies Axioms 1 and 2.
invented entities (1)
-
Double-redundancy function I^{{1}{2}→{1}{2}}_∂
Cite this review
Pith. "Pith review of Beyond integrated information: A taxonomy of information dynamics phenomena." pith.science (2026). https://pith.science/paper/NPRAGIHW
@misc{pith2026190902297,
author = {Pith},
title = {Pith review of: Beyond integrated information: A taxonomy of information dynamics phenomena},
year = {2026},
howpublished = {\url{https://pith.science/paper/NPRAGIHW}},
note = {Machine review of arXiv:1909.02297}
}
abstract
Most information dynamics and statistical causal analysis frameworks rely on the common intuition that causal interactions are intrinsically pairwise -- every 'cause' variable has an associated 'effect' variable, so that a 'causal arrow' can be drawn between them. However, analyses that depict interdependencies as directed graphs fail to discriminate the rich variety of modes of information flow that can coexist within a system. This, in turn, creates problems with attempts to operationalise the concepts of 'dynamical complexity' or `integrated information.' To address this shortcoming, we combine concepts of partial information decomposition and integrated information, and obtain what we call Integrated Information Decomposition, or $\Phi$ID. We show how $\Phi$ID paves the way for more detailed analyses of interdependencies in multivariate time series, and sheds light on collective modes of information dynamics that have not been reported before. Additionally, $\Phi$ID reveals that what is typically referred to as 'integration' is actually an aggregate of several heterogeneous phenomena. Furthermore, $\Phi$ID can be used to formulate new, tailored measures of integrated information, as well as to understand and alleviate the limitations of existing measures.
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Forward citations
Cited by 1 Pith paper
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Reference graph
Works this paper leans on
-
[1]
γ∧∗γ′ :=α∧α′→β∧β′ is a valid meet; and
-
[2]
Note that the fact that ( A,≼) is a lattice implies that α∧β and α∨β are well-defined for all α,β ∈A
γ∨∗γ′ :=α∨α′→β∨β′ is a valid join. Note that the fact that ( A,≼) is a lattice implies that α∧β and α∨β are well-defined for all α,β ∈A . Let us begin with the meet, for which we usem =γ∧∗γ′ as a shorthand notation. First, one can directly check that m≼∗ γ and m≼∗ γ′, given the definition of ≼∗ above and the fact that α∧α′≼ α (and similarly for α′, β, and β...
-
[3]
Additionally, note that I{12}→{1}{2} ∩ = 0, since Red(Y1,Y 2;X1X2) ≤ I(Y2;X1X2) = 0. All this implies that all the redundancies (and hence all the atoms) below{12}→{ 1} are zero, and hence I{12}→{1} ∂ = 1 due to the Moebius inversion formula. • Finally, consider the PPR system where X1,X 2,Y 1 are i.i.d. fair coin flips and Y2 is such that X1⊕ X2 = Y1⊕ Y2....
-
[4]
This implies that all redundancies (and hence atoms) except I{12}→{12} ∩ are zero, and hence using again the Moebius inversion formula I{12}→{12} ∂ = I(X1,X 2;Y1,Y 2) = 1. Appendix E: Results of section ‘Measures of integrated information’ In this appendix we prove the results in Table 1 of the main text, that shows whether each of four measures of integr...
-
[5]
Samplew from a Bernoulli distribution withp = 0.5
-
[6]
• If α ={i}, then xi = w and xj is sampled from a Bernoulli distribution with p = 0.5
Sample x based on α: • If α ={1}{2}, then x1 =x2 =w. • If α ={i}, then xi = w and xj is sampled from a Bernoulli distribution with p = 0.5. • If α ={12}, then x is a random string with parityw
-
[7]
Sample y based on β analogously. In all cases there will be one bit of information ( w) shared between X and Y , hence I(X;Y ) = 1 for any choice of α,β . This can be proven using the fact that for anyα,β , one has H(W ) = 1,H(W|X) =H(W|Y ) = 0, andp(x,y,w ) =p(x|w)p(y|w)p(w). To do so, let us start from the mutual information chain rule: I(X;YW ) =I(X;W ...
-
[8]
J. S. Kelso, Dynamic patterns: The self-organization of brain and behavior (MIT press, 1995)
work page 1995
Show all 50 references
-
[9]
Runge, S
J. Runge, S. Bathiany, E. Bollt, G. Camps-Valls, 11 D. Coumou, E. Deyle, C. Glymour, M. Kretschmer, M. D. Mahecha, J. Mu˜ noz-Mar´ ı,et al., Nature Communications 10, 2553 (2019)
2019
-
[10]
Dosi and A
G. Dosi and A. Roventini, J Evol Econ 29 (2019)
2019
-
[11]
S. L. Bressler and A. K. Seth, Neuroimage 58, 323 (2011)
2011
-
[12]
Tononi, O
G. Tononi, O. Sporns, and G. Edelman, Proceedings of the National Academy of Sciences 91, 5033 (1994)
1994
-
[13]
Balduzzi and G
D. Balduzzi and G. Tononi, PLoS Computational Biology 4, 1 (2008)
2008
-
[14]
Oizumi, L
M. Oizumi, L. Albantakis, and G. Tononi, PLoS Compu- tational Biology 10 (2014), 10.1371/journal.pcbi.1003588
2014 doi
-
[15]
A. K. Seth, A. B. Barrett, and L. Barnett, Philosophi- cal Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 369, 3748 (2011)
2011
-
[16]
A.-M. v. C. van Walsum, Y. Pijnenburg, H. Berendse, B. van Dijk, D. Knol, P. Scheltens, and C. Stam, Clinical Neurophysiology 114, 1034 (2003)
2003
-
[17]
Mediano, A
P. Mediano, A. Seth, and A. Barrett, Entropy 21, 17 (2019)
2019
-
[18]
Quantifying high-order interdependencies via multivariate extensions of the mutual information,
F. Rosas, P. A. M. Mediano, M. Gastpar, and H. J. Jensen, “Quantifying high-order interdependencies via multivariate extensions of the mutual information,” (2019), arXiv:1902.11239 [cs.IT]
2019 arXiv
-
[19]
R. G. James, N. Barnett, and J. P. Crutchfield, Physical Review Letters 116, 238701 (2016)
2016
-
[20]
Lizier, The local information dynamics of distributed computation in complex systems , Ph.D
J. Lizier, The local information dynamics of distributed computation in complex systems , Ph.D. thesis, University of Sydney (2010)
2010
-
[21]
Nonnegative decomposi- tion of multivariate information,
P. L. Williams and R. D. Beer, “Nonnegative decomposi- tion of multivariate information,” (2010), arXiv:1004.2515 [cs.IT]
2010 arXiv
-
[22]
J. P. Crutchfield and D. P. Feldman, Chaos: An Interdis- ciplinary Journal of Nonlinear Science 13, 25 (2003)
2003
-
[23]
Grassberger, International Journal of Theoretical Physics 25, 907 (1986)
P. Grassberger, International Journal of Theoretical Physics 25, 907 (1986)
1986
-
[24]
J. P. Crutchfield, C. J. Ellison, and J. R. Mahoney, Physical Review Letters 103, 094101 (2009)
2009
-
[25]
We use Y1Y2 as a shorthand notation for the random vector (Y1,Y 2)
-
[26]
[ 26] as a measure of integrated information
Syn(X1,X 2;Y1Y2) was in fact proposed in Ref. [ 26] as a measure of integrated information
-
[27]
We focus on the bivariate case for clarity, although our results hold for any N
In a general N-variable case,A is the set of antichains on the lattice (P({1,...,N}),⊆), discussed in [ 14]. We focus on the bivariate case for clarity, although our results hold for any N
-
[28]
A proof of this is provided in the Appendix
-
[29]
We use the shorthand notation Xα = (Xi1,...,X iK ) for α ={i1,...,i K}
-
[30]
A discussion on this issue can be found in the supplementary material
Note that our framework does not prescribe a particular formula for I{1}{2}→{1}{2} ∂ . A discussion on this issue can be found in the supplementary material
-
[31]
The relation between ΦID and causal emergence [ 41] will be described in a separate publication
-
[32]
A. B. Barrett and A. K. Seth, PLoS Computational Biol- ogy 7, 1 (2011)
2011
-
[33]
A principled infotheoretic ϕ-like measure,
V. Griffith, “A principled infotheoretic ϕ-like measure,” (2014), arXiv:1401.0978 [cs.IT]
2014 arXiv
-
[34]
Oizumi, N
M. Oizumi, N. Tsuchiya, and S.-i. Amari, Proceedings of the National Academy of Sciences 113, 14817 (2016)
2016
-
[35]
R. G. James, J. Emenheiser, and J. P. Crutchfield, Journal of Physics A: Mathematical and Theoretical 52, 014002 (2018)
2018
-
[36]
Note that the original definition of causal density is nor- malised by L(L− 1), and has been proven to be bounded by mutual information [10]
-
[37]
P. L. Williams and R. D. Beer, arXiv preprint arXiv:1102.1507 (2011)
2011 arXiv
-
[38]
D. P. Feldman and J. P. Crutchfield, Physics Letters A 238, 244 (1998)
1998
-
[39]
For example, measures that accurately discriminate be- tween neural configurations corresponding to conscious and unconscious states in a particular experimental paradigm [42]
-
[40]
Intuitively, a causal analysis reveals what the system could do, while a dynamical analysis based on attractor statistics reveals what the system actually does
-
[41]
Pearl and D
J. Pearl and D. Mackenzie, The book of why: The new science of cause and effect (Basic Books, 2018)
2018
-
[42]
Also, note that the maximum entropy distributions em- ployed by some causal integration frameworks are well- defined for discrete Markovian systems, but in general may not always exist [43]
-
[43]
C. A. Charalambides, Enumerative combinatorics (Chap- man and Hall/CRC, 2002)
2002
-
[44]
Crampton and G
J. Crampton and G. Loizou, International Mathematical Journal 1, 223 (2001)
2001
-
[45]
R. Quax, O. Har-Shemesh, and P. Sloot, Entropy 19, 85 (2017)
2017
-
[46]
Rassouli, F
B. Rassouli, F. Rosas, and D. G¨ und¨ uz, in2018 IEEE International Workshop on Information Forensics and Security (WIFS) (IEEE, 2018) pp. 1–7
2018
-
[47]
Rosas, V
F. Rosas, V. Ntranos, C. Ellison, S. Pollin, and M. Ver- helst, Entropy 18, 38 (2016)
2016
-
[48]
A. K. Seth, Artificial life 16, 179 (2010)
2010
-
[49]
A. G. Casali, O. Gosseries, M. Rosanova, M. Boly, S. Sarasso, K. R. Casali, S. Casarotto, M.-A. Bruno, S. Laureys, G. Tononi, and M. Massimini, Science Trans- lational Medicine 5, 198ra105 (2013)
2013
-
[50]
Barrett and P
A. Barrett and P. Mediano, Journal of Consciousness Studies 26, 11 (2019)
2019
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