Functional Information Decomposition (FID)

• Functional Information Decomposition (FID) is an information-theoretic framework that partitions mutual information into unique contributions from individual inputs and a synergistic component.
• It operates on complete functional relationships with a uniform input measure and avoids a separate redundancy term by treating independent inputs as statistically isolated.
• FID has been applied to analyze Boolean circuits, dynamical systems, and machine learning models, providing insights into system organization and emergent behavior.

Searching arXiv for papers explicitly mentioning Functional Information Decomposition and closely related frameworks. Functional Information Decomposition (FID) denotes an information-theoretic program for partitioning the mutual information between multiple inputs and an output into components associated with individual variables and with their joint interactions. In the most explicit formulation, FID is defined on complete functional relationships $f:X\to Y$, with a uniform input measure over the full input domain, so that the total information $I(Y;X)$ is decomposed into a sum of independent information terms $I(Y;X_i)$ and a residual synergistic term (Bohm et al., 23 Sep 2025). In adjacent literature, the same expression has also been used more broadly for functionally interpreting multivariate information decompositions: as a functional reading of Integrated Information Decomposition ($\Phi$ID) for dynamical systems (Mediano et al., 2021), as a decomposition of microstate entropy relative to macroscale behavior via the Distributed Information Bottleneck (Murphy et al., 2023), and as a target-free, system-level analogy motivated by System Information Decomposition (SID) (Lyu et al., 2023). These usages share a common objective: to characterize which parts of a system’s informational organization are attributable to local contributions, shared organization, or irreducibly joint structure.

1. Historical emergence and competing usages

FID did not initially enter the literature as a single universally fixed formalism. Earlier work uses the term primarily as an interpretive extension of existing decomposition frameworks. One line identifies FID with a functional reading of $\Phi$ID, in which atoms of dynamical information are read as functionally meaningful modes such as storage, transfer, copy, erasure, upward causation, and downward causation (Mediano et al., 2021). A second line treats FID as the practical decomposition of microstate entropy into the bits most relevant to a chosen macroscale variable $Y$, implemented through the Distributed Information Bottleneck and optimized by machine learning (Murphy et al., 2023). A third line uses SID as a system-level analogue, suggesting that symmetric atoms such as $\text{Red}(X_1,\dots,X_n)$, $\text{Un}(X_i,X_j)$, $\text{Syn}(X_1,\dots,X_n)$, and $\text{Ext}(X_i)$ can serve as building blocks for a function-relative information decomposition (Lyu et al., 2023).

A formally named, first-principles framework appears later in "Functional Information Decomposition: A First-Principles Approach to Analyzing Functional Relationships" (Bohm et al., 23 Sep 2025). There, FID is introduced as a decomposition that “operat[es] on complete functional relationships rather than statistical correlations,” with the central claim that under complete specification and input independence the total mutual information can be partitioned into independent and synergistic components without a separate redundancy term (Bohm et al., 23 Sep 2025).

This divergence of usage is significant. In some papers, FID is not a distinct mathematical object but a conceptual stance placed on PID, $I(Y;X)$0ID, or SID. In the 2025 formulation, by contrast, FID is itself the formal object of study. This suggests that the term now spans both a specific decomposition and a broader research direction concerned with function-relative informational structure.

2. First-principles FID on complete functions

In the first-principles formulation, the basic setup consists of discrete inputs $I(Y;X)$1, a discrete output $I(Y;X)$2, and a complete function

$I(Y;X)$3

that assigns an output to every possible combination of inputs. The function may be deterministic or probabilistic; in the latter case it is specified by conditional probabilities $I(Y;X)$4 over the entire input domain (Bohm et al., 23 Sep 2025).

The framework assumes a uniform measure over the full domain $I(Y;X)$5. Because the domain is fully enumerated, the input variables are treated as statistically independent: $I(Y;X)$6 On this basis, FID defines the total information

$I(Y;X)$7

the independent information for variable $I(Y;X)$8,

$I(Y;X)$9

and the synergistic information

$I(Y;X_i)$0

The central decomposition is therefore

$I(Y;X_i)$1

The framework also defines the solo-synergy of $I(Y;X_i)$2,

$I(Y;X_i)$3

which can be rewritten as

$I(Y;X_i)$4

and the information loss for variable $I(Y;X_i)$5,

$I(Y;X_i)$6

These quantities are intended to separate what an input contributes alone, what depends on its participation in joint structure, and how much total predictive power is lost when it is removed (Bohm et al., 23 Sep 2025).

A defining theorem of this formulation is that, for a complete functional specification with independent inputs, genuine redundancy is ruled out. The paper states that for any pair $I(Y;X_i)$7 and any nonconstant function $I(Y;X_i)$8, it is impossible to have both $I(Y;X_i)$9 and $\Phi$0. The conclusion is that no nontrivial piece of information about $\Phi$1 can be fully specified by two independent inputs simultaneously, so all single-input information about $\Phi$2 is unique and the residual term can be identified directly with synergy (Bohm et al., 23 Sep 2025).

3. Formal properties and canonical examples

The first-principles paper derives several structural properties. Besides the decomposition formula itself, it shows that the solo-synergy terms satisfy

$\Phi$3

It also states an invariance under grouping of inputs: if a subset of variables is grouped into a composite input, the independent information and solo-synergy contributions of the remaining variables are unchanged (Bohm et al., 23 Sep 2025).

The canonical examples are Boolean and cellular-automaton mappings. For the XOR gate,

$\Phi$4

so the entire decomposition is synergistic: $\Phi$5 For the AND gate,

$\Phi$6

and

$\Phi$7

For the “LED square” mapping, the decomposition yields

$\Phi$8

These examples separate pure synergy, mixed independent-plus-synergistic structure, and purely additive independent structure (Bohm et al., 23 Sep 2025).

A higher-order example is Conway’s Game of Life. Taking the $\Phi$9 neighborhood at time $\Phi$0 as nine binary inputs and the focal cell at time $\Phi$1 as output, the paper reports $\Phi$2 bits and $\Phi$3 bits, with each neighbor contributing $\Phi$4 bits, the focal cell contributing $\Phi$5 bits, and a large synergistic residual $\Phi$6 bits (Bohm et al., 23 Sep 2025). The example is used to show that most of the predictive structure of the update rule is not attributable to any individual cell but to joint neighborhood configurations.

The same framework is extended to probabilistic functions through a transition probability matrix. In the three-input example given in the paper, the output entropy is $\Phi$7 bits, the total information is $\Phi$8 bits, the independent information terms are $\Phi$9, $Y$0, and $Y$1 bits, and the synergy is $Y$2 bits (Bohm et al., 23 Sep 2025). This shows that FID in this sense does not require determinism; it decomposes the informational consequences of a fully specified probabilistic mapping.

4. Functional readings of $Y$3ID, PID, and SID

A distinct literature understands FID as a functionally interpretable use of multivariate decomposition frameworks rather than as the particular no-redundancy formalism above. The most developed of these is the interpretation of Integrated Information Decomposition, $Y$4ID, as a decomposition of functional modes in dynamical systems (Mediano et al., 2021). For a Markovian two-element system, $Y$5ID decomposes the time-delayed mutual information

$Y$6

into atoms indexed by pairs of PID modes,

$Y$7

yielding $Y$8 atoms in the two-variable case (Mediano et al., 2021). These atoms are organized into six qualitative modes of information dynamics: storage, copy, transfer, erasure, downward causation, and upward causation. Within this reading, FID is not another lattice or measure; it is the interpretation of $Y$9ID atoms as functional channels and of groups of atoms as functional modes.

This interpretation leads to explicit decompositions of standard dynamical quantities. For example, active information storage for element 1 is written as

$\text{Red}(X_1,\dots,X_n)$0

while transfer entropy from 1 to 2 becomes

$\text{Red}(X_1,\dots,X_n)$1

This is used to distinguish “genuine directed transfer” from contributions due to shared information or emergent joint structure (Mediano et al., 2021).

SID provides a different, target-free perspective. Instead of decomposing how sources inform a distinguished target, SID treats all variables symmetrically and introduces system-level atoms $\text{Red}(X_1,\dots,X_n)$2, $\text{Red}(X_1,\dots,X_n)$3, $\text{Red}(X_1,\dots,X_n)$4, and $\text{Red}(X_1,\dots,X_n)$5 (Lyu et al., 2023). For three variables it gives the entropy decomposition

$\text{Red}(X_1,\dots,X_n)$6

Although SID does not formally define FID, it motivates the idea that functions $\text{Red}(X_1,\dots,X_n)$7 implemented by a system can be treated as new random variables and then analyzed through symmetric system-level atoms (Lyu et al., 2023).

A further conceptual precursor is the cooperative-game-theoretic decomposition of mutual information into contributions $\text{Red}(X_1,\dots,X_n)$8 indexed by subsets $\text{Red}(X_1,\dots,X_n)$9, with no explicit redundancy term (Ay et al., 2019). The decomposition satisfies non-negativity, completeness, symmetry, a singleton property analogous to identity, and additivity across independent subsystems (Ay et al., 2019). This framework does not use the term FID, but it offers a subset-indexed decomposition that, like the 2025 first-principles FID, can be read as emphasizing unique and synergy-like contributions without requiring a separate redundancy atom.

5. Machine-learning and algebraic implementations

Another major strand treats FID as a scalable, function-relative decomposition problem rather than as a closed-form lattice decomposition. In "Information decomposition in complex systems via machine learning," the objective is to identify, bit by bit, which microscopic degrees of freedom are most relevant to a macroscopic variable $\text{Un}(X_i,X_j)$0 (Murphy et al., 2023). The system is described by a microstate vector

$\text{Un}(X_i,X_j)$1

and the method uses the Distributed Information Bottleneck with separate lossy compressions $\text{Un}(X_i,X_j)$2 for each component: $\text{Un}(X_i,X_j)$3 By sweeping $\text{Un}(X_i,X_j)$4, one obtains a family of solutions that trade total extracted information $\text{Un}(X_i,X_j)$5 against predictive information $\text{Un}(X_i,X_j)$6, visualized in a “distributed information plane” (Murphy et al., 2023). In this usage, FID means a decomposition of microstate entropy into those per-component bits that are most predictive of a chosen macroscale behavior.

The implementation is variational. The compression term uses an upper bound

$\text{Un}(X_i,X_j)$7

and the predictive term is estimated by a lower bound through cross-entropy on $\text{Un}(X_i,X_j)$8. For interpretive analysis, the paper refines the estimates of $\text{Un}(X_i,X_j)$9 using InfoNCE lower bounds and leave-one-out upper bounds, with intervals reported as about $\text{Syn}(X_1,\dots,X_n)$0 bits (Murphy et al., 2023). The emphasis here is practical tractability for tens to hundreds of variables rather than exact decomposition into PID-style atoms.

A complementary implementation concern arises in PID and $\text{Syn}(X_1,\dots,X_n)$1ID themselves. "The Fast Möbius Transform: An algebraic approach to information decomposition" provides an algebraic method for computing PID and $\text{Syn}(X_1,\dots,X_n)$2ID atoms efficiently through Möbius inversion on redundancy lattices (Jansma et al., 2024). The paper identifies the PID redundancy lattice as a free distributive lattice and gives a closed-form Möbius function

$\text{Syn}(X_1,\dots,X_n)$3

together with the product form for $\text{Syn}(X_1,\dots,X_n)$4ID,

$\text{Syn}(X_1,\dots,X_n)$5

This reduces the computation of atoms to sparse matrix–vector multiplication and makes full decompositions practical up to five sources (Jansma et al., 2024). Although the paper does not formally define FID, it explicitly presents the algebraic and computational framework as adaptable to “functional-level” decompositions in which the basic elements are functional units such as frequency bands, modules, or other mechanisms (Jansma et al., 2024).

6. Empirical applications and scientific significance

Empirical usage of FID-style ideas spans logic-gate models, physiological coupling, brain dynamics, and neuronal-network topology. Under the $\text{Syn}(X_1,\dots,X_n)$6ID reading, three systems can share the same whole-minus-sum integrated information $\text{Syn}(X_1,\dots,X_n)$7 while having distinct functional organizations: a “copy transfer” system dominated by $\text{Syn}(X_1,\dots,X_n)$8, a “downward XOR” system dominated by $\text{Syn}(X_1,\dots,X_n)$9, and a “parity-preserving random” system dominated by $\text{Ext}(X_i)$0 (Mediano et al., 2021). The comparison is used to show that scalar integrated-information measures can collapse functionally distinct mechanisms into the same value.

The same paper analyzes a whole-brain Dynamic Mean Field model. As coupling $\text{Ext}(X_i)$1 increases, the firing rate jumps near $\text{Ext}(X_i)$2; $\text{Ext}(X_i)$3 becomes negative because it is dominated by redundancy; and the revised $\text{Ext}(X_i)$4, which focuses on synergy plus transfer, peaks near $\text{Ext}(X_i)$5, a regime described as consistent with awake conscious brain activity (Mediano et al., 2021). A further case study on heart–respiration coupling from the Fantasia dataset reports that $\text{Ext}(X_i)$6, but $\text{Ext}(X_i)$7ID attributes the asymmetry largely to $\text{Ext}(X_i)$8 and redundancy-related terms rather than to a difference in the genuine transfer atom $\text{Ext}(X_i)$9 (Mediano et al., 2021). The implication is that apparent directional asymmetries in transfer entropy may reflect emergent joint dynamics rather than direct signaling.

The machine-learning decomposition via DIB is illustrated on a Boolean circuit with ten binary inputs and on amorphous materials undergoing plastic deformation (Murphy et al., 2023). In the Boolean-circuit example, the DIB trajectory closely traces the upper envelope of the $I(Y;X)$00 discrete subsets of inputs, identifying the most informative subsets without exhaustive search (Murphy et al., 2023). In the glass example, approximately $I(Y;X)$01 bit of radial-density information already yields about $I(Y;X)$02 accuracy for gradual quench and $I(Y;X)$03 for rapid quench, while around $I(Y;X)$04 bits give about $I(Y;X)$05 and $I(Y;X)$06 accuracy, respectively (Murphy et al., 2023). The selected bits concentrate in small radii near the center and align with features of the radial distribution functions $I(Y;X)$07 and $I(Y;X)$08, indicating which structural variations are most related to imminent rearrangement (Murphy et al., 2023).

In neuronal-network analysis, $I(Y;X)$09ID has been used to decompose transfer entropy between neurons or regions into unique, redundant, and synergistic transfer. In "Integrated Information Decomposition Unveils Major Structural Traits of $I(Y;X)$10 and $I(Y;X)$11 Neuronal Networks," the authors report that the unique information transfer term $I(Y;X)$12 is the most relevant measure for uncovering structural topological details from activity data, whereas redundant information mainly introduces residual information for that task (Menesse et al., 2024). In simulations, the unique-transfer atom gives the best reconstruction at low false positive rates, and a critical threshold $I(Y;X)$13 is identified as an optimal trade-off point in ROC analysis (Menesse et al., 2024). In vitro, applying the method to calcium-imaged cortical cultures on anisotropic substrates reveals modularity with average $I(Y;X)$14, global efficiency $I(Y;X)$15, mean in-degree $I(Y;X)$16, and connection-angle peaks at $I(Y;X)$17, reflecting the direction of the imprinted bands (Menesse et al., 2024).

These studies show that FID, whether understood as a formal function-based decomposition or as a functional interpretation of $I(Y;X)$18ID or DIB, is used to move beyond scalar information summaries. It provides either exact or approximate profiles of which components, pathways, or configurations are responsible for prediction, coordination, or emergent organization.

7. Conceptual controversies, limitations, and open problems

The principal conceptual controversy concerns redundancy. The first-principles FID paper argues that under complete functional specification and independent inputs, redundancy is mathematically impossible and should not appear as a separate component (Bohm et al., 23 Sep 2025). This stands in direct contrast to PID-based and SID-based traditions, where redundancy is a central category. In $I(Y;X)$19ID, for example, redundancy is one of the four basic PID modes used to build dynamical atoms (Mediano et al., 2021), and in SID redundancy is a symmetric system-level quantity $I(Y;X)$20 that enters entropy, total-correlation, and co-information decompositions (Lyu et al., 2023). The difference is rooted in assumptions: the 2025 FID treats information structure as a property of a fully specified function under independent inputs, whereas PID, $I(Y;X)$21ID, and SID work directly with observational or system-level joint distributions, where input dependencies and shared informational structure are generally present.

A second limitation is completeness. The first-principles FID explicitly states that incomplete truth tables do not determine a unique decomposition: the partial specification

$I(Y;X)$22

is compatible with XOR, OR, and probabilistic mixtures, each with different synergy and independent-information values (Bohm et al., 23 Sep 2025). The paper therefore proposes sampling over possible completions to obtain ranges or distributions of decompositions rather than a single value. This approach is epistemically conservative, but it also shows that FID in this sense depends critically on how the functional model is specified.

A third issue is scalability. Exact multivariate information decomposition is computationally difficult. PID and $I(Y;X)$23ID suffer from the growth of the redundancy lattice with the number of variables; DIB avoids exhaustive combinatorics but replaces exact decomposition with optimization over learned compressions (Murphy et al., 2023). The Fast Möbius Transform improves tractability for PID and $I(Y;X)$24ID up to about five sources (Jansma et al., 2024), but high-dimensional exact decompositions remain difficult.

A fourth issue is estimator and model dependence. The machine-learning approach requires expressive encoders and predictors, sufficient data, and reliable mutual-information bounds (Murphy et al., 2023). $I(Y;X)$25ID applications depend on a redundancy function, often the minimal mutual information (MMI) redundancy, which is described as tending to overestimate redundancy and therefore to provide lower bounds on unique and synergistic contributions (Menesse et al., 2024). SID likewise remains agnostic about a universally agreed redundancy functional and gives explicit formulas mainly for three-variable systems (Lyu et al., 2023).

A plausible implication is that the long-term trajectory of the field will not be a single universal FID formalism, but a family of decompositions matched to distinct epistemic settings. When the system is a known input–output function, the first-principles decomposition supplies direct formulas for independent and synergistic information (Bohm et al., 23 Sep 2025). When the system is an observed stochastic dynamical process, $I(Y;X)$26ID supplies a dynamical taxonomy of functional modes (Mediano et al., 2021). When the objective is scalable discovery of behaviorally relevant structure in high-dimensional data, the DIB-based approach provides a practically computable bit-by-bit decomposition of microstate information (Murphy et al., 2023). Across these settings, the unifying idea is not a single equation but a methodological commitment: information should be decomposed in a way that respects the functional roles played by parts, coalitions, and emergent patterns within the system.