Explicit Formula for Partial Information Decomposition in Multivariate Systems

Core Concepts

This paper proposes an explicit formula for calculating the unique, redundant, and synergistic information atoms in a partial information decomposition of multivariate systems, which satisfies all the previously proposed axioms and properties.

Abstract

The paper addresses the problem of extending Shannon's mutual information to multivariate systems, which fails to capture the fine-grained interactions between the source and target variables. In 2010, Williams and Beer proposed the concept of Partial Information Decomposition (PID) to decompose the mutual information into unique, redundant, and synergistic information atoms. However, despite extensive efforts, a general formula satisfying all the proposed axioms and properties has not been found. The key contributions of this paper are: Introduction of the "do-operation", which adjusts the marginal distribution of the target variable while minimally impacting its connections with other variables. This operation is inspired by Judea Pearl's do-calculus in causal analysis. Defining the unique information as the expectation of mutual information between the adjusted target variable and the source variable, given the other source variable(s). This definition satisfies all the proposed axioms and properties. Deriving the definitions of the redundant and synergistic information atoms from the unique information definition, using the relationships established in the axioms. Providing intuitive explanations for the role of the do-operation and the interpretation of the information atoms. The paper proves that the proposed definitions satisfy all the axioms and properties, including nonnegativity, commutativity, additivity, and continuity. The unique information is shown to be bounded by the mutual information and the conditional entropy, and the synergistic information is nonnegative under the closed-system assumption.

Stats

None.

Quotes

None.

Key Insights Distilled From

Explicit Formula for Partial Information Decomposition

by Aobo Lyu,And... at arxiv.org 05-07-2024

https://arxiv.org/pdf/2402.03554.pdf

Deeper Inquiries

What are the potential applications of the proposed partial information decomposition formula beyond the examples mentioned in the paper

The proposed partial information decomposition formula has a wide range of potential applications beyond the examples mentioned in the paper. One key application could be in the field of network analysis, particularly in understanding complex interactions between nodes in a network. By decomposing the mutual information into unique, redundant, and synergistic components, the formula can provide insights into how different nodes influence each other in a network setting. This can be valuable in optimizing network structures, identifying key nodes for information flow, and understanding emergent properties in complex networks. Another application could be in the study of biological systems, such as gene regulatory networks. By applying the partial information decomposition formula to gene expression data, researchers can uncover the unique, redundant, and synergistic interactions between genes, leading to a better understanding of gene regulatory mechanisms, disease pathways, and potential therapeutic targets. Furthermore, the formula could be utilized in machine learning and artificial intelligence applications, particularly in feature selection and model interpretability. By decomposing the information shared between input features and output predictions, the formula can help identify the most relevant features, reduce redundancy in the data, and highlight synergistic effects that contribute to model performance. This can lead to more efficient and interpretable machine learning models.

How can the interpretation of the synergistic information atom be further explored, especially in cases where it may take negative values

The interpretation of the synergistic information atom, especially in cases where it may take negative values, opens up interesting avenues for further exploration. In scenarios where the synergistic information atom is negative, it could indicate competitive interactions or conflicting influences between the source variables on the target variable. This negative synergistic information may suggest that the combined effect of the source variables is less than what would be expected based on their individual contributions, highlighting complex relationships and dependencies in the system. Further exploration of negative synergistic information could involve studying the dynamics of the system under different conditions or perturbations to understand how the interactions between variables change. It could also involve investigating the impact of feedback loops, nonlinear relationships, or hidden variables that may contribute to negative synergistic effects. By delving deeper into the implications of negative synergistic information, researchers can gain valuable insights into the underlying mechanisms driving complex systems.

Can the do-operation and the proposed approach be extended to handle more than two source variables, and what challenges might arise in such generalizations

The do-operation and the proposed approach can indeed be extended to handle more than two source variables, although challenges may arise in such generalizations. Extending the approach to multiple source variables would involve adapting the formula to account for the interactions between multiple sources and the target variable. This extension would require defining unique, redundant, and synergistic information atoms for each combination of source variables and the target variable, leading to a more comprehensive decomposition of mutual information in multivariate systems. Challenges in generalizing the approach to multiple source variables may include increased computational complexity, as the number of possible combinations and interactions grows exponentially with the number of sources. Ensuring the consistency and interpretability of the information atoms across multiple variables would also be a challenge, as the interactions between multiple sources can be more intricate and difficult to disentangle. Additionally, validating the extended approach with real-world data and ensuring its applicability to diverse domains would be crucial in overcoming challenges and establishing the effectiveness of the generalized formula.