# Partial Effective Information Decomposition for Synergistic Causality

arXiv: https://arxiv.org/abs/2605.03267 · HTML: https://mlsanity.com/paper/2605.03267/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 52 references that are in the map.
- First posted on arXiv 5 May 2026.

## Builds on (most cited first, 19 of 19)

- [Foundation Models for Time Series Analysis: A Tutorial and Survey](https://mlsanity.com/paper/2403.14735/) (2024)
- [Deep Time Series Models: A Comprehensive Survey and Benchmark](https://mlsanity.com/paper/2407.13278/) (2024)
- [Causal Consistency of Structural Equation Models](https://mlsanity.com/paper/1707.00819/) (2017)
- [Granger Causality: A Review and Recent Advances](https://mlsanity.com/paper/2105.02675/) (2021)
- [On the representation and learning of monotone triangular transport maps](https://mlsanity.com/paper/2009.10303/) (2020)
- [Optimal Training of Mean Variance Estimation Neural Networks](https://mlsanity.com/paper/2302.08875/) (2023)
- [When the map is better than the territory](https://mlsanity.com/paper/1612.09592/) (2016)
- [Reconciling emergences: An information-theoretic approach to identify causal emergence in multivariate data](https://mlsanity.com/paper/2004.08220/) (2020)
- [In reply to Faes et al. and Barnett et al. regarding "A study of problems encountered in Granger causality analysis from a neuroscience perspective"](https://mlsanity.com/paper/1709.10248/) (2017)
- [Measuring Integrated Information: Comparison of Candidate Measures in Theory and Simulation](https://mlsanity.com/paper/1806.09373/) (2018)
- [Quantifying Consistency and Information Loss for Causal Abstraction Learning](https://mlsanity.com/paper/2305.04357/) (2023)
- [Greater than the parts: A review of the information decomposition approach to causal emergence](https://mlsanity.com/paper/2111.06518/) (2021)
- [PCDCNet: A Surrogate Model for Air Quality Forecasting with Physical-Chemical Dynamics and Constraints](https://mlsanity.com/paper/2505.19842/) (2025)
- [Finding emergence in data by maximizing effective information](https://mlsanity.com/paper/2308.09952/) (2023)
- [Measuring integrated information from the decoding perspective](https://mlsanity.com/paper/1505.04368/) (2015)
- [Neural Information Squeezer for Causal Emergence](https://mlsanity.com/paper/2201.10154/) (2022)
- [Emergence and Causality in Complex Systems: A Survey on Causal Emergence and Related Quantitative Studies](https://mlsanity.com/paper/2312.16815/) (2023)
- [Dissecting Spectral Granger Causality through Partial Information Decomposition](https://mlsanity.com/paper/2603.07634/) (2026)
- [Higher order definition of causality by optimally conditioned transfer entropy](https://mlsanity.com/paper/2409.08295/) (2024)

## Abstract (arXiv)

Causality is a central topic in scientific inquiry, yet for complex systems, the identification and analysis of synergistic causation remain a challenging and fundamental problem. In the context of causal relations among multivariate variables, a decomposition framework grounded in interventionist causation is still lacking. To address this gap, this paper proposes Partial Effective Information Decomposition (PEID), a framework that decomposes the influence of multiple source variables on a target variable under maximum-entropy interventions into unique and synergistic information, thereby providing a unified and computable characterization of synergistic causal relations. Theoretically, in the three-variable case, the proposed framework is compatible with the major axioms of Partial Information Decomposition (PID). Empirically, under maximum-entropy interventions, correlations among input variables are removed, causing redundancy to vanish and thereby enabling PEID to compute synergistic relations. Furthermore, based on this framework, it is possible to define causal graphs containing hyperedges as well as downward causation, thus offering a unified toolkit for analyzing cross-scale and multivariate causal mechanisms in complex systems. Finally, applying the framework to a machine-learning-based air quality forecasting task on KnowAir-V2, we demonstrate that PEID can extract interpretable inter-station causal structures from a learned dynamical model. These results suggest that PEID provides a general interventionist information-theoretic tool for analyzing multivariate and synergistic causal mechanisms in complex systems.