Closed-Form Gaussian Estimators for Multi-Source Partial Information Decomposition
2026-05-11 • Information Theory
Information Theory
AI summaryⓘ
The authors developed new mathematical formulas to measure how different continuous data sources share and combine information when there are more than two sources involved. Their method works specifically for Gaussian data and gives exact calculations instead of relying on learning algorithms. They tested their formulas to show they are reliable, fast, and stable with limited data. This work extends previous methods that were limited to just two sources and provides a first closed-form tool for analyzing multi-source continuous data.
partial information decompositionGaussian estimatormulti-source informationsynergyredundancyconditional independencecovariance matrixlog-determinantaffine invariancepermutation symmetry
Authors
Aobo Lyu, Andrew Clark, Netanel Raviv
Abstract
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.