Functional Information Decomposition: A First-Principles Approach to Analyzing Functional Relationships

TL;DR

Functional Information Decomposition is introduced, both a computational and theoretical framework, which defines informational components with respect to a system's complete input-output mapping, thereby addressing a core cross-scale inference problem: determining how information carried by individual components combines to shape system-level behavior.

Abstract

A central challenge in analyzing multivariate interactions within complex systems is to decompose how multiple inputs jointly determine an output. Existing approaches generally operate on observed probability distributions and can conflate a system's intrinsic functional logic with statistical artifacts of limited data. As a result, distinct systems can yield identical observations, rendering information decomposition fundamentally underdetermined and obscuring true higher-order interactions. We introduce Functional Information Decomposition (FID), both a computational and theoretical framework, which defines informational components with respect to a system's complete input-output mapping, thereby addressing a core cross-scale inference problem: determining how information carried by individual components combines to shape system-level behavior. When the mapping is fully specified, FID provides a unique decomposition into independent and synergistic contributions. Crucially, given only partial observations, FID characterizes the entire space of consistent decompositions by sampling compatible functions, making inferential limits explicit. A complementary geometric perspective clarifies the structural origin of informational components. We demonstrate FID's interdisciplinary utility on canonical logical functions, Conway's Game of Life, and gene-expression-based prediction of cancer drug response, and provide an open-source implementation. By separating functional architecture from observational distribution, FID offers a principled foundation for analyzing multivariate dependence in both fully and partially observed complex systems.

Figures (4)

• Figure 1: OR--XOR completion space. The total information and synergy for completions for a partially observed function compatible with OR or XOR. Black markers indicate deterministic completions (solid dot: OR; open circle: XOR). Red indicates probabilistic completions. The dashed lines intersect at the maximum-entropy (“neutral”) completion, where the missing input condition $X_1 = 1, X_2 = 1$, is equally likely to yield $Y = 0$ or $Y = 1$.
• Figure 2: FID decomposition for a seven-gene predictor of resistance to the apoptosis inhibitor ABT737. Each panel shows decompositions for functional specifications consistent with the observed genetic data, obtained by sampling. Black points indicate decompositions resulting from fully deterministic completions. Colored points correspond to decompositions of probabilistic completions, with the color indicating the Dirichlet concentration parameter (green: more deterministic; red: more probabilistic). Rows correspond to individual genes, and columns show different projections for each gene, as labeled. Dashed lines indicate values obtained from the maximum entropy assumption.
• Figure 3: Coverage types illustrated. Left (Injective): Each input value (X) maps to a unique output value (Y). Center (Pseudo-injective): Each input value maps to a distinct, non-overlapping subset of outputs—supports remain disjoint. Right (Overjective): Input supports overlap; the split-colored (red/yellow) output indicates that multiple input values can produce the same output, creating fundamental ambiguity.
• Figure 4: In the LED Square example(left), both inputs, S1 and S2, are pseudo-injective, while the function as a whole is injective. In contrast, XOR (right) is overjective in both the input and at the function scopes.