bibkey: barrett2015gaussianinformation authors: Adam B. Barrett year: 2015 title: Exploration of synergistic and redundant information sharing in static and dynamical Gaussian systems doi: null url: https://arxiv.org/abs/1411.2832v2 claim: Sections 2–3 give finite jointly Gaussian conditional covariance, entropy and mutual information, and the whole-minus-sum information difference for three scalar Gaussian variables. strata_touched: [] license: citation-only triage: anchor

Joint Gaussian conditional information and net information difference

The primary text is arXiv:1411.2832v2, §§2–3. Section 2, equations (5)–(6), gives the Schur conditional covariance and the observation-dependent conditional mean. Equations (7)–(14) give differential entropy, the information chain rule, and the Gaussian determinant formulas. The continuous variables have densities with respect to Lebesgue measure; the displayed inverses and finite logarithms require nonsingular covariances.

Section 3, equations (15)–(20), treats a positive-definite three-variable correlation matrix with off-diagonal entries $a,b,c$. Its whole-minus-sum quantity is

$$\begin{array}{rl}\mathrm{WMS}(X;Y,Z)& =I(X;Y,Z)-I(X;Y)-I(X;Z)\\ & =I(X;Y\mid Z)-I(X;Y)\\ & =\frac{1}{2}\log \frac{(1-a^2)(1-b^2)(1-c^2)}{1-a^2-b^2-c^2+2abc}.\end{array}$$

This difference is net synergy minus redundancy in the paper’s terminology; it does not by itself specify all parts of a partial information decomposition.

The legal parity-source application, Theorem 28.4 consumes this formula with $X=Y_A$, $Y=Y_B$, $Z=Y_C$ and $\mathcal{U}=-\vartheta \mathrm{WMS}$. Its independent calibrated noises, common Gaussian source, rank-one interventions and source-to-control map are additional declared data. The paper supplies the information identity, not the weighted five-edge determinant polynomial or the legal parity-source target. No article text, figure or supplementary material is reproduced here.