arXiv subjects

Aobo Lyu

Publications and source records attributed to Aobo Lyu.

7 recordsLinked to original sources

arXiv2026

Closed-Form Gaussian Estimators for Multi-Source Partial Information Decomposition

Computing multi-source partial information decomposition (PID) for continuous data is hard: existing closed-form Gaussian estimators are restricted to two source variables, while continuous arbitrary-source estimators are typically learning-based and do not provide closed-form expressions. To address this, we develop closed-form Gaussian estimators for multi-source PID. We provide two-source redundancy, multi-source unique information, the K-th order synergistic effect from source subsets of size K, and the total synergistic effect. The estimators are derived from the conditional-independence-based information measures introduced in our earlier work, under which every quantity reduces to a log-determinant expression in covariance blocks of the system. The resulting estimator is plug-in consistent, affine invariant, source-permutation symmetric, and additive over independent systems. We validate it on a controlled Gaussian benchmark, evaluate its computational efficiency against baselines, and confirm its numerical stability in finite-sample regimes. To our knowledge, this is the first covariance-based closed-form estimator that provides multi-source continuous PID measures.

cs.IT↗

arXiv2026

Structural Impossibility of Antichain-Lattice Partial Information Decomposition

Partial Information Decomposition (PID) represents multivariate mutual information via antichain-lattice that aims to specify which source groups can recover which informational components of a target. For three or more sources, widely desired PID axioms become mutually incompatible. This is often treated as an axiomatic tuning issue. This paper argues that the obstruction is representational, rooted in the antichain indexing itself, so that purely axiomatic adjustments within an antichain-lattice structure cannot resolve it in general. We first introduce System Information Decomposition (SID) for the special target-free three-variable setting, obtaining a self-consistent entropy decomposition with an operational redundancy definition. More fundamentally, we then show that for general multivariate PID, there is no universal rule that recovers the decomposed mutual information from the antichain-indexed information atoms. In particular, two systems can share identical atoms regardless of any axioms while having different mutual information. These results reveal the limits of antichain-lattice and motivate relation-based foundations for multivariate information measures.

cs.IT↗

arXiv2025

The Whole Is Less than the Sum of Parts: Subsystem Inconsistency in Partial Information Decomposition

Partial Information Decomposition (PID) was proposed by Williams and Beer in 2010 as a tool for analyzing fine-grained interactions between multiple random variables, and has since found numerous applications ranging from neuroscience to privacy. However, a unified theoretical framework remains elusive due to key conceptual and technical challenges. We identify and illustrate a crucial problem: PID violates the set-theoretic principle that the whole equals the sum of its parts (WESP). Through a counterexample in a three-variable system, we demonstrate how such violations naturally arise, revealing a fundamental limitation of current lattice-based PID frameworks. To address this issue, we introduce a new axiomatic framework, termed System Information Decomposition (SID), specifically tailored for three-variable systems. SID resolves the WESP violation by redefining the summation rules of decomposed information atoms based on synergistic relationships. However, we further show that for systems with four or more variables, no partial summation approach within the existing lattice-based structures can fully eliminate WESP inconsistencies. Our results thus highlight the inherent inadequacy of (antichain) lattice-based decompositions for general multivariate systems.

cs.IT↗

arXiv2025

Multivariate Partial Information Decomposition: Constructions, Inconsistencies, and Alternative Measures

While mutual information effectively quantifies dependence between two variables, it does not by itself reveal the complex, fine-grained interactions among variables, i.e., how multiple sources contribute redundantly, uniquely, or synergistically to a target in multivariate settings. The Partial Information Decomposition (PID) framework was introduced to address this by decomposing the mutual information between a set of source variables and a target variable into fine-grained information atoms such as redundant, unique, and synergistic components. In this work, we review the axiomatic system and desired properties of the PID framework and make three main contributions. First, we resolve the two-source PID case by providing explicit closed-form formulas for all information atoms that satisfy the full set of axioms and desirable properties. Second, we prove that for three or more sources, PID suffers from fundamental inconsistencies: we review the known three-variable counterexample where the sum of atoms exceeds the total information, and extend it to a comprehensive impossibility theorem showing that no lattice-based decomposition can be consistent for all subsets when the number of sources exceeds three. Finally, we deviate from the PID lattice approach to avoid its inconsistencies, and present explicit measures of multivariate unique and synergistic information. Our proposed measures, which rely on new systems of random variables that eliminate higher-order dependencies, satisfy key axioms such as additivity and continuity, provide a robust theoretical explanation of high-order relations, and show strong numerical performance in comprehensive experiments on the Ising model. Our findings highlight the need for a new framework for studying multivariate information decomposition.

cs.IT↗

arXiv2024

Emergence and Causality in Complex Systems: A Survey on Causal Emergence and Related Quantitative Studies

Emergence and causality are two fundamental concepts for understanding complex systems. They are interconnected. On one hand, emergence refers to the phenomenon where macroscopic properties cannot be solely attributed to the cause of individual properties. On the other hand, causality can exhibit emergence, meaning that new causal laws may arise as we increase the level of abstraction. Causal emergence theory aims to bridge these two concepts and even employs measures of causality to quantify emergence. This paper provides a comprehensive review of recent advancements in quantitative theories and applications of causal emergence. Two key problems are addressed: quantifying causal emergence and identifying it in data. Addressing the latter requires the use of machine learning techniques, thus establishing a connection between causal emergence and artificial intelligence. We highlighted that the architectures used for identifying causal emergence are shared by causal representation learning, causal model abstraction, and world model-based reinforcement learning. Consequently, progress in any of these areas can benefit the others. Potential applications and future perspectives are also discussed in the final section of the review.

physics.soc-ph↗

arXiv2024

Explicit Formula for Partial Information Decomposition

Mutual information between two random variables is a well-studied notion, whose understanding is fairly complete. Mutual information between one random variable and a pair of other random variables, however, is a far more involved notion. Specifically, Shannon's mutual information does not capture fine-grained interactions between those three variables, resulting in limited insights in complex systems. To capture these fine-grained interactions, in 2010 Williams and Beer proposed to decompose this mutual information to information atoms, called unique, redundant, and synergistic, and proposed several operational axioms that these atoms must satisfy. In spite of numerous efforts, a general formula which satisfies these axioms has yet to be found. Inspired by Judea Pearl's do-calculus, we resolve this open problem by introducing the do-operation, an operation over the variable system which sets a certain marginal to a desired value, which is distinct from any existing approaches. Using this operation, we provide the first explicit formula for calculating the information atoms so that Williams and Beer's axioms are satisfied, as well as additional properties from subsequent studies in the field.

cs.IT↗

arXiv2023

System Information Decomposition

To characterize the complex higher-order interactions among variables within a system, this study introduces a novel framework, termed System Information Decomposition (SID), aimed at decomposing the information entropy of variables into information atoms based on their interrelations. Diverging from the established Partial Information Decomposition (PID) framework, which predominantly concentrates on the directional interactions stemming from an array of source variables to a single target variable, SID adopts a holistic approach, scrutinizing the interactions across all variables within the system. Specifically, we proved all the information atoms are symmetric, which means the disentanglement of unique, redundant, and synergistic information from any specific target variable. Hence, our proposed SID framework can capture the symmetric pairwise and higher-order relationships among variables. This advance positions SID as a promising framework with the potential to foster a deeper understanding of higher-order relationships within complex systems across disciplines.

cs.IT↗