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Measuring Integrated Information: Comparison of Candidate Measures in Theory and Simulation

Pedro A. M. Mediano, Anil K. Seth, Adam B. Barrett

arXiv:1806.09373v1q-bio.NC

TL;DR

A theoretically principled universal measure of dynamical complexity remains elusive, and candidate measures differ in their constructions. The paper compares their theoretical properties and behavior on non-trivial networks; only a subset best captures conjoined integration and segregation in the tested setting.

  • Problem

    A theoretically principled universal measure of dynamical complexity remains elusive, and differences in candidate-measure behavior are difficult to interpret when their constructions vary.

    Source: §1 · p. 2, §3.2 · p. 7

  • Method

    The paper compares candidate measures by their theoretical construction and behavior on non-trivial network architectures.

    Source: §5 · p. 25

  • Results

    Among the measures considered, ψ, Φ*, and CD best capture conjoined segregation and integration on small networks with Gaussian linear AR dynamics.

    Source: §5 · p. 25

  • Takeaways & Limitations

    The results motivate continued development of ψ, Φ*, and CD as theoretically sound and empirically adequate measures of integrated information.

    Source: §5 · p. 25

    Because the measures use empirical rather than maximum-entropy distributions, they measure dynamics rather than mechanism, warranting caution about treating them as generalisations or approximations of IIT Φ measures.

    Source: §5.3 · p. 27

Abstract

from arXiv · show

Integrated Information Theory (IIT) is a prominent theory of consciousness that has at its centre measures that quantify the extent to which a system generates more information than the sum of its parts. While several candidate measures of integrated information (`$\Phi$') now exist, little is known about how they compare, especially in terms of their behaviour on non-trivial network models. In this article we provide clear and intuitive descriptions of six distinct candidate measures. We then explore the properties of each of these measures in simulation on networks consisting of eight interacting nodes, animated with Gaussian linear autoregressive dynamics. We find a striking diversity in the behaviour of these measures -- no two measures show consistent agreement across all analyses. Further, only a subset of the measures appear to genuinely reflect some form of dynamical complexity, in the sense of simultaneous segregation and integration between system components. Our results help guide the operationalisation of IIT and advance the development of measures of integrated information that may have more general applicability.

1 Introduction

Dynamical complexity involves a balance between integration and segregation, but a theoretically principled universal measure remains elusive. The paper explains six candidate measures and compares their behavior on non-trivial network models.

1 Introduction

  • IIT measures quantify how much information the whole system generates beyond the sum of its parts, with the partition chosen across the weakest links.The paper treats these measures as operationalisations of dynamical complexity rather than focusing on their proposed connection to consciousness.Source: §1 · p. 2
  • The study compares candidate measures theoretically and on networks to assess their practical utility as operationalisations of integrated information and dynamical complexity.It examines how topology, coupling strength, and noise-input correlation affect the measures, and compares them with global correlation.Source: §1 · p. 2
  • The review covers five integrated-information measures and causal density, which sums independent information transfers without using a minimum information partition.The five measures are Φ, ˜Φ, ψ, Φ*, and ΦG.Source: §1 · p. 2
  • The simulations use eight-node networks with varied architectures and noisy vector autoregressive dynamics to compare measure behavior.The analyses also relate each measure to global correlation as a simple dynamical control.Source: §1 · p. 2
  • Appendix A.1 derives new Gaussian-system formulae for Φ*, correcting the previous formulae in Ref. [35].Other appendices provide further derivations of mathematical properties of the measures.Source: §1 · p. 3

2 Notation, convention and preliminaries

The preliminaries establish notation and assumptions for variables, system partitions, and time series, then introduce entropy and information quantities used to define the candidate measures.

2 Notation, convention and preliminaries

  • A partition divides a system into non-overlapping, non-empty parts; variables may be continuous or discrete, and the paper assumes stationary, ergodic systems.Time-series variables are indexed by subscripts, and each system variable is denoted Xi.Source: §2 · pp. 3, 5
  • Entropy quantifies uncertainty, while differential entropy extends the definition to continuous variables with densities but is not invariant to rescaling.The paper assumes continuous variables have densities with respect to Lebesgue measure.Source: §2 · p. 3
  • The Kullback-Leibler divergence measures distributional dissimilarity and is a non-symmetric distance used in information geometry..Source: §2 · p. 4
  • Mutual information measures interdependence, is symmetric, and is bounded by the entropy of either variable.It is invariant under injective transformations and can track temporal information preservation through I(Xt; Xt−τ).Source: §2 · p. 4
  • For Gaussian variables, the paper uses covariance and cross-covariance matrices and the conditional covariance formula.Under the stationary assumption, the state covariance is unchanged across time lags.Source: §2 · pp. 4–5

3 Integrated information measures

This section reviews candidate integrated-information measures and how they operationalize information beyond a partition using spontaneous information dynamics, including their information functionals, partition schemes, interpretations, assumptions, and practical limitations. It also describes causal-density calculation procedures for different variable types and situates the other measures relative to Φ-3.0 and a broader taxonomy of candidates.

3.1 Overview

  • The measures reviewed, except for causal density, were inspired by Balduzzi and Tononi’s Φ2008, which uses information about a hypothetical maximum-entropy past and is practically limited to discrete Markovian systems.The measures considered here instead use spontaneous information dynamics, making them well-defined for stochastic systems with a well-defined Lebesgue measure and estimable from empirical distributions when stationarity can be assumed.Source: §3.1 · p. 5

3.2 Minimum information partition

  • The minimum information partition identifies the weakest link by minimizing normalized effective information, but measures use differing partition rules that confound comparisons.Φ and ˜Φ share a partition scheme, ψ minimizes unnormalized effective information, and Φ∗, ΦG, and CD use the atomic partition; the paper’s results instead minimize unnormalized measures over even-sized bipartitions.Source: §3.2 · pp. 6–7

3.3 Whole-minus-sum integrated information Φ

  • The whole-minus-sum Φ measures the loss in time-delayed mutual information after partitioning, but can be negative and is not an absolute measure of integration.Under spontaneous dynamics, two expressions equivalent under uniform perturbation become distinct, equally valid analogs of Φ2008; positive Φ suffices to infer the whole predicts the future better, while nonpositive values do not establish the opposite and can indicate net redundancy [9].Source: §3.3 · pp. 7–8

3.4 Integrated stochastic interaction ˜Φ

  • Integrated stochastic interaction ˜Φ measures increased uncertainty about the past when a system is split and, unlike Φ, is guaranteed to be non-negative.It uses an uncertainty-increase definition of effective information rather than Φ’s loss-of-information formulation.Source: §3.4 · p. 9

3.5 Integrated synergy ψ

  • Integrated synergy ψ measures information about the system’s future that the parts provide exclusively synergistically, but its operational definition depends on a contested redundancy measure.For the paper’s linear-Gaussian systems, the authors use minimum mutual information PID [9].Source: §3.5 · pp. 10–11

3.6 Decoder-based integrated information Φ∗

  • Decoder-based Φ∗ measures predictive information lost when a receiver decodes with a restricted model that ignores correlations between partitions.It uses mismatched decoding information and lacks an interpretable formulation in simpler functionals such as entropy or mutual information; for discrete and linear-Gaussian systems, its optimization has an analytic closed form when the joint probability table or covariance is known.Source: §3.6 · pp. 12–13

3.7 Geometric integrated information ΦG

  • Geometric ΦG compares the full system model with a restricted model that severs links between parts while allowing correlations across them.It has no known closed-form solution; discrete and linear-Gaussian cases can be estimated through constrained multivariate optimization with a differentiable, convex objective [14].Source: §3.7 · pp. 13–14

3.8 Causal density

  • Causal density vanishes in purely segregated or purely integrated systems, is non-negative, and is bounded above by total time-delayed mutual information.Unlike the other candidates, it sums information transfers rather than directly measuring how much the whole exceeds its parts; it is defined as average pairwise conditioned transfer entropy.Source: §3.8 · pp. 14–15
  • Causal density calculation covers discrete and continuous linear-Gaussian variables, while arbitrary continuous distributions require nearest-neighbour methods.Source: §3.8 · p. 15

3.9 Other measures

  • Φ-3.0 follows the conceptually distinct latest IIT, is more computationally expensive, and is not adapted here to spontaneous information dynamics.The other reviewed measures, except causal density, were inspired by Φ2008.Source: §3.9 · pp. 15–16
  • Tegmark’s taxonomy [41] overlaps with the reviewed measures and includes minor variants; φMkk′ is state-resolved transfer entropy is covered through causal density, and φnpk is undefined for continuous variables.ΦG and ψ fall outside the taxonomy.Source: §3.9 · p. 16

4 Results

The section compares six integrated-information measures on Gaussian autoregressive networks, varying connection strength, topology, density, and noise correlation. Their responses differ, and several measures show patterns consistent with capturing both integration and segregation.

4 Results

  • The simulations use stable Gaussian linear AR processes and compare measures across connection strength, topology, density, and noise correlation.Stationary covariance is obtained from a discrete-time Lyapunov equation; results minimize unnormalised measures over even-sized bipartitions.Source: §4 · pp. 16–17, §4.1 · p. 17

4.2 Two-node network

  • For the two-node process, ψ, Φ*, and CD decrease to zero as noise obscures the effect of coupling, whereas Φ can become negative.The system is non-stationary when a ≥ 0.5, so the coupling sweep evaluates values below that limit.Source: §4.2 · pp. 18–19, Figure 2 · p. 18
  • In the two-node model, ψ, Φ*, and CD decrease with noise correlation and increase with coupling strength, unlike several other measures.TDMI and ΦG are unaffected by noise correlation, while ˜Φ and average correlation increase with both coupling and noise correlation.Source: §4.2 · pp. 18–19, Figure 3 · p. 19

4.3 Eight-node networks

  • In the eight-node Φ-optimal network, ˜Φ diverges at high noise correlation, while ψ and ΦG rise slightly and Φ∗ and CD decline less than in simpler systems.Φ does not show the instability and negative values seen in the two-node model.Source: §4.3 · pp. 19–20, Figure 4 · p. 20
  • Across the six eight-node topologies, the unidirectional ring is most complex for all measures except ˜Φ, while the fully connected network is least complex.The fully connected network’s low scores are attributed to its large node correlations.Source: §4.3 · p. 21, Figure 5 · p. 21, Figure 6 · p. 22
  • Across network topologies, TDMI, ΦG, Φ*, and ψ largely agree on rankings, but those rankings differ from small-world-index rankings.The Spearman correlation between small-world-index rankings and those of TDMI, ΦG, Φ*, and ψ is around −0.4.Source: §4.3 · pp. 21–23, Table 3 · p. 22

4.4 Random networks

  • For random networks, all measures except ΦG show high values at intermediate density; beyond that, fixed spectral radius weakens individual connections.ΦG rises markedly with density, while its response to noise correlation is moderate; the other measures divide in their response to noise correlation.Source: §4.4 · pp. 23–24, Figure 7 · p. 23
  • Φ*, ψ, ΦG, and CD show intermediate peaks against average correlation, while ˜Φ trends upward and Φ trends downward.The sampled networks are relatively uniformly distributed along the average-correlation axis, so the peaks are not sampling artefacts.Source: Figure 8 · p. 24, §4.4 · pp. 24–25

5 Discussion

Across Gaussian linear AR networks, the measures differ substantially: ψ, Φ∗, and CD best capture joint segregation and integration on small networks. The discussion also identifies limits from partition search, Gaussian assumptions, and using empirical rather than maximum-entropy distributions.

5 Discussion

  • ψ, Φ∗, and CD best capture conjoined segregation and integration on small networks with Gaussian linear AR dynamics.They rise with coupling strength, fall with increasing noise-input correlation, and score highest at intermediate connection counts and average component correlations.Source: §5 · p. 25
  • At fixed overall coupling strength, ring structures usually yield the highest scores, while none of the measures strongly reflects graph-theoretic network complexity.˜Φ tracks noise-input correlation more than coupling strength; ΦG mainly depends on coupling strength; Φ behaves erratically.Source: §5 · p. 25
  • The authors motivate continued development of ψ, Φ∗, and CD as theoretically sound and empirically adequate integrated-information measures.Their summary also notes that the measures have distinct mathematical properties and varied empirical behavior.Source: §5 · p. 25

5.1 Partition selection

  • The comparison holds partition selection constant and searches even-sized bipartitions, avoiding normalization across differently sized partitions.Uneven partitions require normalization factors that can introduce instability under continuous parameter changes and numerical error.Source: §5.1 · p. 26
  • Searching all partitions is more comprehensive but takes O(nn)9, versus O(2n) for bipartitions and O(n2) for even bipartitions.The authors describe a trade-off between computation time and coverage, and discuss restricted-search approximations and an optimization method limited to instantaneous mutual information.Source: §5.1 · p. 26

5.2 Continuous variables and the linear Gaussian assumption

  • The comparison is limited to continuous Gaussian states; the authors call for tests on discrete and non-Gaussian continuous systems.General continuous probability densities are difficult to estimate, whereas the studied AR systems permit analytical information-theoretic calculations.Source: §5.2 · pp. 26–27

5.3 Empirical as opposed to maximum entropy distribution

  • Using the empirical rather than maximum-entropy distribution makes the measures about dynamics, not mechanisms, and requires identifying stationary epochs.Long data segments may not have constant statistics, while short segments may not explore all possible states; the authors also caution against treating these measures as generalizations or approximations of IIT’s fundamental Φ measures.Source: §5.3 · p. 27

6 Final remarks

The paper reviews candidate integrated-information measures and compares them on simulated networks with Gaussian dynamics. Its findings show substantial behavioral differences even among measures with similar theoretical properties, while assessing dynamical complexity remains an open challenge.

6 Final remarks

  • The paper reviews candidate integrated-information measures and compares them on simulated data from Gaussian dynamics across network topologies.The authors frame satisfactory informational measurement of dynamical complexity as necessary for further IIT development and empirical application.Source: §6 · p. 27
  • Measures with similar theoretical properties can behave substantially differently, even on simple systems, motivating empirical investigation of their behavior.The review identifies promising theoretical approaches for designing adequate measures.Source: §6 · p. 28
  • Assessing dynamical complexity, integrated information, or co-existing integration and segregation remains an important outstanding challenge.The authors note its relevance to consciousness theories and more generally to settings where local and global dynamics relate.Source: §6 · p. 28

A.1 Derivation of I∗in Gaussian systems

The appendix derives a closed-form expression for mismatched decoding information in a Gaussian dynamical system. It defines the distributions involved, works through the integral terms, and reports numerical verification of a formula that differs from an earlier one.

A.1 Derivation of I∗in Gaussian systems

  • The appendix provides a closed-form expression for mismatched decoding information in a Gaussian dynamical system.The formula is written as a function of β for a stationary continuous random process.Source: §A.1 · p. 28
  • The derivation uses p(x) for Xt, p(x, x̃) for (Xt, Xt−τ), and q(x|x̃) for the conditional distribution under the partition.For Gaussian Xt with covariance ΣX and zero mean, the appendix proceeds to derive the expression.Source: §A.1 · p. 28
  • The derivation evaluates the integral in parts and recombines the terms to obtain the formula.The text identifies the first and second terms before putting them together.Source: §A.1 · pp. 29–30
  • Numerical methods verified the formula for ˜I(β), which differs from the formula reported by Oizumi et al. [35].Source: §A.1 · p. 30

A.2 ˜I(β) is concave in β in Gaussian systems

The appendix proves that ˜I(β) is concave in β for Gaussian systems. It uses the expression’s component terms and standard concavity-preserving properties to establish the result.

A.2 ˜I(β) is concave in β in Gaussian systems

  • The proof aims to establish concavity of ˜I(β), which gives it a unique maximum and allows standard convex-optimization tools.The appendix states the optimization consequence of concavity.Source: §A.2 · p. 30
  • One term in the expression is linear in β, while the log-sum-exp component is convex and the minus sign makes the resulting term concave.The appendix uses this structure in its concavity argument.Source: §A.2 · pp. 30–31
  • Affine combinations and non-negative weighted sums preserve concavity, so the proof reduces to showing concavity of a component term.The outer integral preserves concavity because p(x) > 0.Source: §A.2 · p. 31

B Bounds on causal density

The section uses the mutual-information chain rule and non-negativity to prove that causal density is bounded above by time-delayed mutual information.

B Bounds on causal density

  • Causal density is upper-bounded by time-delayed mutual information, a condition identified as a fundamental requirement for an integrated-information measure [37].The proof rewrites causal density in terms of mutual information, applies the chain rule and non-negativity, and sums over all n(n −1) pairs.Source: B Bounds on causal density · pp. 31–32

C Properties of integrated information measures

The section establishes properties of candidate integrated-information measures using mutual-information identities, while showing that their properties differ. In particular, one candidate is not bounded by time-delayed mutual information, whereas another is.

C Properties of integrated information measures

Integrated synergy ψ

The properties examined for integrated synergy and decoder-based integrated information include non-negativity, rescaling invariance, bounds by TDMI, and time symmetry. The reported proofs and counterexamples are specific to the formulations used in the paper.

Integrated synergy ψ

  • The integrated synergy ψ is not time-symmetric in the cited AR-system counterexample; the proof applies only to the MMI-PID formulation used in the paper.The properties table also reports ψ as non-negative, rescaling-invariant, and bounded by TDMI.Source: Integrated synergy ψ · p. 33

Decoder-based integrated information Φ∗

  • Decoder-based integrated information Φ∗ is non-negative, rescaling-invariant, and bounded by TDMI according to the listed proofs.For rescaling, the argument uses a diagonal matrix of positive real values and states that the factors cancel.Source: Decoder-based integrated information Φ∗ · p. 33

Geometric integrated information ΦG

The geometric measure ΦG is reported as non-negative, rescaling-invariant, time-symmetric, and bounded by TDMI. The causal-density properties listed are non-negativity, rescaling invariance, and a TDMI bound, but not time symmetry.

Geometric integrated information ΦG

  • ΦG is non-negative and time-symmetric, and its rescaling invariance follows because rescaling preserves the relevant KL divergence between the full distribution and its M-projection.Its bound ΦG ≤ I(Xt; Xt−τ) follows because the manifold used to define TDMI is a subset of the restricted-model manifold Q.Source: Geometric integrated information ΦG · pp. 33–34

Causal density

  • Causal density is non-negative, rescaling-invariant, and bounded by TDMI; its time asymmetry follows from the non-symmetry of transfer entropy.Non-negativity follows by rewriting causal density as a sum of conditional mutual-information terms.Source: Causal density · p. 34
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