Decomposing Multivariate Information Rates in Networks of Random Processes

Daniele Marinazzo; Gorana Mijatovic; Laura Sparacino; Leonardo Ricci; Luca Faes; Sebastiano Stramaglia; Yuri Antonacci

REVIEW 3 major objections 4 minor 1 cited by

Decomposing Multivariate Information Rates in Networks of Random Processes

T0 review · 3 major / 4 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims that the mutual information rate between a target random process and a set of source processes can be decomposed over the same redundancy lattice as PID into non-negative unique, redundant, and synergistic…

desk verdict PIRD is a genuine extension of PID to time series; the lattice formalism and M=2 simulations are solid, but the claimed non-negative decomposition fails for M>2 because SMMI inherits MMI's negative-atom problem. read the letter →

arxiv 2502.04555 v3 pith:HZF6PMUJ submitted 2025-02-06 stat.ME cs.ITmath.IT

Laura Sparacino , Gorana Mijatovic , Yuri Antonacci , Leonardo Ricci , Daniele Marinazzo , Sebastiano Stramaglia , Luca Faes This is my paper · ORCID

classification stat.MEcs.ITmath.IT MSC 62M1094A1560G1062H20

keywords partialinformationdecompositionmutualratespectralredundancymultivariatetimeseriesnetworkphysiologyGaussianprocessestransferentropyfrequency-domainanalysis

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 introduces the partial information rate decomposition (PIRD), a framework that extends partial information decomposition (PID) from random variables to random processes. It replaces the mutual information between a target variable and source variables with the mutual information rate between a target process and source processes, and runs the same redundancy lattice of PID over this rate. The load-bearing new object is a spectral redundancy rate function: at each frequency, redundancy is the minimum of the spectral MIRs between the target and each source group, and the global redundancy rate is its integral over frequency. The paper argues that this yields a non-negative decomposition into unique, redundant, and synergistic information rates, that it equals static PID when the processes are memoryless, and that it equals PID applied to transfer entropy when the coupling is strictly causal with no instantaneous or reverse interactions. If correct, analysts of multivariate time series no longer need to pretend the samples are independent and identically distributed.

What carries the argument

The carried object is the spectral minimum mutual-information (SMMI) redundancy rate, defined in Eq. (23) as the minimum, at each frequency $\omega$, of the spectral MIRs between the target process and the source groups composing an atom. For jointly Gaussian processes, each spectral MIR is given by the determinant formula $\frac{1}{2}\log\frac{|P_X(\omega)|P_Y(\omega)}{|P_{[YX]}(\omega)|}$, so every atom is computed from determinants of sub-blocks of the power spectral density matrix; the PSD itself is estimated by fitting a vector autoregressive model and applying spectral factorization. This machinery lets redundancy be defined locally in frequency, where the spectral MIR is non-negative, and then integrated to produce time-domain information-rate atoms, while also permitting band-limited decompositions restricted to specific oscillatory components.

What would settle it

Take a stationary nonlinear vector process with a known mutual information rate, for example a bivariate process generated by a nonlinear coupling whose exact transfer entropy can be computed analytically, and compare the PIRD atoms computed through the Gaussian VAR route against a consistent model-free estimate of the same spectral MIR atoms; if the two sets of atoms disagree substantially or become negative, the Gaussian spectral implementation is not a faithful estimator of the general MIR decomposition.

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Extended reading notes

Core claim

The central claim is that the information a target process $Y$ shares per unit time with source processes $X_1,\ldots,X_M$ splits, without loss or double-counting, into atoms indexed by the redundancy lattice, exactly as static PID does for random variables. The split is achieved by defining a redundancy rate as a pointwise minimum of spectral MIRs: for each atom, the redundancy rate is the integral over frequency of the minimum, over the atom's elements, of the spectral MIR between the target and that element. For jointly Gaussian processes, each spectral MIR is computed from determinants of sub-blocks of the power spectral density matrix, and the paper shows that the spectral redundancy rate satisfies the standard PID axioms, that the frequency-domain and time-domain decompositions commute with integration, and that the method collapses to the zero-lag PID and to the PID of the joint transfer entropy in the two limiting dynamic regimes. The framework is therefore claimed to resolve the mismatch between PID's implicit memorylessness and the temporal correlations present in real network data, and to do so while preserving a non-negative, interpretable decomposition.

Load-bearing premise

The practical version of the framework assumes the analyzed vector process is stationary and jointly Gaussian, so that the spectral MIR equals the determinant formula and the spectrum can be estimated from a finite-order VAR model; real physiological series may violate these conditions, and no model-free estimator of the decomposition is provided.

Editorial extensions

If this is right

Temporal correlations no longer disqualify a dataset from information decomposition: the MIR-based atoms remain non-negative and are well defined for processes with memory, unlike the zero-lag PID applied to dependent samples.
Frequency-band integration makes the decomposition scale-specific: unique, redundant, and synergistic information rates can be reported within physiologically meaningful bands such as low-frequency and high-frequency ranges, revealing higher-order interactions that may cancel out in whole-band averages.
The framework subsumes two existing practices: it reduces to the static PID in the memoryless case and to the PID of transfer entropy in the strictly causal case, so previous results can be reinterpreted as boundary cases of the same lattice construction.
In the physiological application, the finding that cerebrovascular and cardiovascular interactions are predominantly redundant and that redundancy increases with postural stress provides a spectrally resolved, target-specific readout of high-order coupling in network physiology.
The equivalence between frequency-domain and time-domain PIRD means that researchers can choose either representation freely, or combine them, without changing the resulting information-rate atoms.

Reading between the lines

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

Beyond the paper: because the provided implementation is parametric and Gaussian, a natural testable extension is a model-free estimator of the spectral MIR and of the SMMI redundancy rate, which would show whether the atoms change materially when gaussianity or linearity fails.
Beyond the paper: the authors note that spectral pointwise minimization is less conservative than time-domain MMI redundancy; if that ordering holds more generally, PIRD may reduce the known overestimation of redundancy in MMI-style schemes and make redundancy rates more comparable across different redundancy functions.
Beyond the paper: the decomposition treats the chosen target and the specified sources only, so a hidden common driver outside the source set could inflate the redundant or synergistic atoms; conditioning on exogenous processes or embedding PIRD in a graphical model is a plausible next step that the paper does not develop.
Beyond the paper: the band-limited formulation suggests a direct application to any rhythmic dataset, such as neural oscillations or cardiorespiratory coupling, where whole-band information measures may obscure the coexistence of synergy in one band and redundancy in another.

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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

To be candid: the core idea of this paper—swap MI for mutual information rate in the PID lattice, define a spectral minimum-MI redundancy, and recover static PID and TE-PID as limit cases—is genuinely new and should be taken seriously. The M=2 simulations are clean and the spectral decomposition is a nice trick. The appendices prove what they claim: the SMMI redundancy function satisfies the four Williams–Beer axioms.

The load-bearing problem is that the paper also claims a non-negative decomposition of the MIR. That does not hold for more than two sources. In the memoryless Gaussian limit the SMMI redundancy coincides exactly with the standard MMI redundancy (the paper proves this in A.1). MMI is known to produce negative partial information atoms for M≥3. A concrete example: three memoryless Gaussians with X1=X2, X3 independent of everything, Y=X1+N. Then R=min(I1,I2,I3)=0, U1=I1, U2=I2, and the coarse-grained synergy S=I−U1−U2−R≈−I2<0. Same thing happens with the spectral PI rates. Since the paper applies the M=3 version to the physiological network, this is not a minor technicality; it breaks the stated 'non-negative decomposition' promise.

What else is soft: no code or data are released, the Gaussian/VAR assumptions are not validated on the physiological time series (only the model is fitted), and the simulation validation is self-referential—the behaviors are built in and then recovered. The paper honestly notes that redundancy choice is axiomatic, so that alone is not a flaw, but it weakens the claim that SMMI is the right choice.

What survives: the lattice formalism and Möbius inversion are correct, the spectral integration equivalence (A.2) is fine, and the M=2 behavior is exactly what you want. The idea is worth publishing, but the non-negativity claim needs to be either proven under restricted conditions or removed. A revision that adds an explicit caveat about M≥3 and provides code would be credible.

I'd send this to a serious referee. The idea is important, the formalism is mostly sound, and the flaw is a specific overclaim, not a collapse. But I would not cite it as a validated tool until the non-negativity issue is resolved.

Referee Report

3 major / 4 minor

Summary. The paper proposes a Partial Information Rate Decomposition (PIRD) that extends the Williams-Beer partial information decomposition from random variables to stationary random processes, replacing mutual information by mutual information rate and using a newly defined spectral minimum-MI (SMMI) redundancy rate. The decomposition is implemented for Gaussian processes through the power spectral density and VAR models, and it is applied to simulated VAR networks and to a physiological network with three source processes. The paper claims that PIRD yields non-negative unique, redundant, and synergistic information-rate contributions, captures full temporal and spectral structure, and reduces to static PID in the memoryless case and to PID of transfer entropy in the strictly causal case.

Significance. The conceptual step of decomposing mutual information rate over the same redundancy lattice as PID is natural and potentially useful, and the frequency-domain formulation offers a computationally tractable, spectrally resolved decomposition for Gaussian VAR processes. The appendix proofs (A.1-A.3) give useful support for the basic properties of the SMMI redundancy rate. However, the central non-negativity claim fails for more than two sources, and some of the reduction claims are overstated. The M=3 physiological application therefore rests on an unsupported property of the proposed redundancy function.

major comments (3)

[Frequency-domain PIRD, Eqs. (23)-(25), (15)] The claimed non-negative decomposition is false for M>2. In the memoryless Gaussian limit the SMMI redundancy rate coincides with the MMI redundancy rate (Appendix A.1). Take M=3 with Y=X1+N, X1=X2, X3 independent of Y, X1, X2, and N independent small Gaussian noise. Then I(Y;X3)=0, so R_{Y;X}=min(I_{Y;X1}, I_{Y;X2}, I_{Y;X3})=0, and Eq. (15) gives S_{Y;X}=I_{Y;X1,X2,X3}-U_{Y;X1}-U_{Y;X2}-U_{Y;X3}-R_{Y;X}=I-I-0-0=-I<0. For the full lattice, take X1=X2=X3=Y+N_i with i.i.d. noises so all individual MIs are equal to some finite I; Eq. (25) gives PI({1})=I-PI({1}{2}{3})-PI({1}{2})-PI({1}{3})=I-I-I-I=-2I<0. Since the M=3 physiological analysis uses exactly this redundancy, the non-negativity claim and the interpretation of the M=3 atoms are unsupported. The authors should either restrict non-negativity claims and applications to M=2, impose additional constraints to enforce non-negativity, or adopt a redundancy function known to yield non-negative atoms.
[Partial Information Rate Decomposition, Eq. (16)] The statement that in the strictly causal case 'the PIRD reduces to a PID applied to the TE' is only true at the level of the total quantity. The PIRD is solved with the SMMI redundancy rate (23)-(24), whereas the PID of the transfer entropy as implemented in the paper uses a time-domain MMI redundancy (17). Because the integral of a minimum of spectra is not equal to the minimum of the integrals (Appendix A.1 only proves an inequality), the PIRD atoms and the TE-based MMI atoms will generally differ in the presence of self-dependencies. This should be stated explicitly, or the comparison in Section 'Effects of changes in the network topology' should be framed as comparing two different redundancy schemes rather than as PIRD 'reducing' to TE-PID.
[Theoretical Examples, first paragraph] The simulations are not an independent validation of the decomposition: the VAR parameters are chosen to produce the behaviors that PIRD then recovers, as the text itself says ('Simulations are designed to induce expected behaviors and thus aim to provide a validation'). This demonstrates internal consistency with the chosen axioms but does not test whether SMMI is a correct or interpretable redundancy function. An independent check for M=3 would require a setting with known, non-negative ground-truth atoms, or at least a direct stress test of the negative-atom issue described above.

minor comments (4)

[Application to Physiological Networks] There are several typos that should be corrected: 'th spontaneous variability' should be 'the spontaneous variability'; 'operalization' should be 'operationalization'; 'strightforward' should be 'straightforward'; 'withing' should be 'within'.
[Figure 5 caption] The caption refers to 'the parameters c1, c2', but the text and the simulation use a single parameter c; this should be corrected.
[Frequency-domain PIRD] The term 'pointwise' is used for frequency-specific redundancy. This should be carefully distinguished from pointwise (realization-specific) local mutual information in the PID literature to avoid confusion.
[Formulation for Gaussian processes] The discussion of fitting a VAR model as 'assuming a linear model rather than a linear process' is useful, but the actual condition for Eqs. (28)-(29) is joint Gaussianity and stationarity; the Wold-decomposition argument alone does not justify the Gaussian assumption, and this should be clarified.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PIRD construction is definitional in the standard PID sense, and the new SMMI redundancy rate is an independent choice whose Möbius inversion is a mathematical identity.

full rationale

The paper's derivation chain is self-contained. The PIRD is built on the standard Williams–Beer lattice: given any redundancy function, the partial information rates are uniquely obtained by Möbius inversion (Eqs. 13–14 and 25). This is a mathematical identity of the lattice framework, not a circular reduction, because the proposed SMMI redundancy rate (Eqs. 23–24) is defined independently of the atoms, as a frequency-wise minimum of spectral MIRs. The comparison I_SMMI <= I_MMI (Appendix A.1), the time–frequency equivalence (Appendix A.2), and the Williams–Beer axioms (Appendix A.3) are proven from stated definitions and standard inequalities; they do not presuppose the target decomposition. The simulations are explicitly designed as demonstrations that the measures behave as expected under controlled VAR parameters, not as predictions fitted to data; no parameter is fitted to a subset of outcomes and then renamed a prediction. The physiological application fits a VAR model from data, but the PIRD values are then computed from the fitted model; this is an estimation pipeline, not a circular validation. Self-citations appear in implementation details (VAR identification tool) and in the physiological interpretation section, but the central mathematical derivation does not rest on them. The possible failure of non-negativity of coarse-grained atoms for M > 2 with MMI-type redundancy is a correctness concern about the chosen redundancy function, not a circularity of the derivation. Overall, no load-bearing step reduces to its own inputs.

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

PIRD is a definitional extension of PID: once the SMMI redundancy rate (Eq. 23) is fixed, the atoms are determined by Möbius inversion. No parameters are fitted to data in the method itself; the main burdens are the axiomatic choice of redundancy and the stationary-Gaussian/VAR assumptions needed for computation. No new physical entities are introduced.

assumptions (6)

standard math Williams-Beer redundancy lattice and consistency equations apply verbatim to mutual information rate.
Sections 'Partial Information Rate Decomposition' and 'Frequency-domain PIRD' reuse the lattice and Möbius inversion of PID (Eqs. 13-14, 19-21).
ad hoc to paper Minimum spectral MIR is a valid redundancy rate for a set of sources.
Eq. (23) defines redundancy as the pointwise frequency-domain minimum of spectral MIRs; the paper acknowledges that no canonical redundancy definition exists and that MMI-based measures tend to overestimate redundancy.
domain assumption Processes are stationary and jointly Gaussian for the spectral expansion of MIR.
Eqs. (27)-(29) express spectral MIR via PSD determinants, valid for jointly Gaussian processes; the section 'Formulation for Gaussian processes' states this limitation.
domain assumption A finite-order VAR model properly represents the dynamics.
Eq. (31) fits a VAR(p) model to estimate the PSD; the paper invokes Wold's theorem but notes that nonlinear processes may require infinite order.
domain assumption Coarse-grained PID aggregation (kth-order) transfers to information rates.
The section 'Partial Information Rate Decomposition' extends the M=3 coarse-grained PID of [18] to PIRD atoms; this is assumed rather than proven for rates.
standard math Frequency-specific spectral MIR obeys the chain rule for groups of sources.
Used in Appendix A.3 (Proposition A.3.3) to show subset equality; follows from the Gaussian spectral representation, but is not explicitly proved.

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Cite this review

Pith. "Pith review of Decomposing Multivariate Information Rates in Networks of Random Processes." pith.science (2026). https://pith.science/paper/HZF6PMUJ

@misc{pith2026250204555,
  author       = {Pith},
  title        = {Pith review of: Decomposing Multivariate Information Rates in Networks of Random Processes},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HZF6PMUJ}},
  note         = {Machine review of arXiv:2502.04555}
}

read the original abstract

The Partial Information Decomposition (PID) framework has emerged as a powerful tool for analyzing high-order interdependencies in complex network systems. However, its application to dynamic processes remains challenging due to the implicit assumption of memorylessness, which often falls in real-world scenarios. In this work, we introduce the framework of Partial Information Rate Decomposition (PIRD) that extends PID to random processes with temporal correlations. By leveraging mutual information rate (MIR) instead of mutual information (MI), our approach decomposes the dynamic information shared by multivariate random processes into unique, redundant, and synergistic contributions obtained aggregating information rate atoms in a principled manner. To solve PIRD, we define a pointwise redundancy rate function based on the minimum MI principle applied locally in the frequency-domain representation of the processes. The framework is validated in benchmark simulations of Gaussian systems, demonstrating its advantages over traditional PID in capturing temporal correlations and showing how the spectral representation may reveal scale-specific higher-order interactions that are obscured in the time domain. Furthermore, we apply PIRD to a physiological network comprising cerebrovascular and cardiovascular variables, revealing frequency-dependent redundant information exchange during a protocol of postural stress. Our results highlight the necessity of accounting for the full temporal statistical structure and spectral content of vector random processes to meaningfully perform information decomposition in network systems with dynamic behavior such as those typically encountered in neuroscience and physiology.

Figures

Figures reproduced from arXiv: 2502.04555 by the authors.

**Figure 1.** Standard and coarse-grained partial information decomposition, superimposed on the redundancy lattices for M sources. (a) Standard PID on the redundancy lattice for 3 variables (M = 2). The alphabet of source combinations is A = {{1}{2}, {1}, {2}, {12}}, where {1} denotes S1 and {2} denotes S2. The MI between the target and the set of sources is decomposed into a redundant (magenta), a synergistic (blue) and two uni… view at source ↗

**Figure 2.** [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** Example of how the spectral redundancy rate [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The presence of temporal correlations has a profound impact on the multivariate information shared at lag zero by multiple random processes. (a) Simulation design, where Y is the target process and {X1, X2} is the group of sources; time-lagged interactions (solid black…

**Figure 5.** Figure 5: Changes in the network topology have a profound impact on the multivariate information shared by multiple random processes. a) Simulation design, where Y is the target process and {X1, X2} is the group of sources; time-lagged interactions (solid black arrows) are set …

**Figure 6.** Figure 6: Coexistence of redundant and synergistic characters of interactions in different spectral bands elicited by frequency-specific PIRD. (a) Network structure, with Y receiving from X1, oscillating at 0.3 Hz, and X3, oscillating at 0.1 Hz, and X1 sending to X2, oscillating…

**Figure 7.** Figure 7: Coarse-grained PIRD applied to the physiological network of mean cerebral blood flow velocity (F), mean arterial pressure (M), heart period (H) and respiration (R) assessed in patients prone to develop postural-related syncope. The mutual (left column: IF ;H (black dot…

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Partial Information Rate Decomposition
stat.ME 2025-02 conditional novelty 7.0 of 10

The paper defines Partial Information Rate Decomposition, a spectral lattice method that decomposes mutual information rate for stationary Gaussian processes into redundant, unique, and synergistic dynamic components.

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