Can Gaussian BROJA Partial Information Decomposition Be Computed in Closed Form?

A closed form replaces iterative coupling search

BROJA partial information decomposition splits the information two sources X and Y carry about a target M into redundant, unique, and synergistic parts. Its definition rests on an assumption that unique and redundant information depend only on the two source–target marginals, not on the full joint distribution. That turns the problem into a search over compatible couplings: joint distributions that keep both marginals fixed while changing how the sources depend on each other.

Earlier work showed that when each source–target pair is jointly Gaussian, this search can be restricted to Gaussian couplings and reduced to a finite-dimensional numerical optimization. The remaining question, which this paper answers, was whether that population minimum has a closed form. The authors show it does, for any number of target dimensions, provided the joint covariance matrices of (M, X) and (M, Y) are positive definite.

Posterior covariances define the feasible set

The key objects are the conditional covariances Γ_X = Cov(M|X) and Γ_Y = Cov(M|Y), computed as Schur complements of the joint covariance matrices. Under the Gaussian assumption these are constant matrices, and each describes how much uncertainty about M remains after seeing one source.

The optimization is then written over posterior-error covariance matrices Γ that are no larger than both Γ_X and Γ_Y in the Loewner order, meaning Γ_Y − Γ is positive semidefinite and likewise for Γ_X. The paper's target is the Γ in that feasible family with the largest determinant. Maximizing this determinant is equivalent to minimizing the union information, the part of the total information about M that is not synergistic.

Generalized eigenvalues decide each target direction

The solution diagonalizes the problem. The authors take the generalized eigenvalues λ_i of Γ_Y with respect to Γ_X, obtained from the eigendecomposition of Γ_X^(−1/2) Γ_Y Γ_X^(−1/2). In these coordinates the two posterior covariances become the identity and a diagonal matrix Λ.

The optimal Γ★ keeps, in each direction, the smaller of the two posterior variances: its diagonal entries are min{1, λ_i}. The paper proves this optimizer exists, is unique, and is feasible. The proof uses Hadamard's determinant inequality, and uniqueness follows because equality in that inequality forces the transformed matrix to be diagonal. In plain terms, the best coupling never lets the target remain more uncertain than either source alone would allow, direction by direction.

Union and unique information in closed form

With Γ★ in hand, the BROJA union information is ½ log₂(det Σ_M / det Γ★), where Σ_M is the covariance of the target. The unique information of each source is a sum over the same eigenvalues: UI_X = ½ Σ_i [log₂ λ_i]₊ and UI_Y = ½ Σ_i [−log₂ λ_i]₊, with [a]₊ = max{a, 0}.

These two quantities answer different questions and should not be conflated. Union information is the non-synergistic portion of the total information about M; unique information is what one source contributes that the other lacks. The paper shows unique information is determined directly by the generalized eigenvalues of the two posterior covariance matrices, which is why the computation reduces to an eigendecomposition rather than a search.

Agreement with numerical solvers to numerical error

The authors implemented the closed form as Eigen-PID in Python and compared it with two optimization-based Gaussian PID methods, Thin-PID and Tilde-PID, on the gain-sweep benchmark from Venkatesh et al. (2023) using the setting of Zhao et al. (2025). Across PID components, differences between the three methods stayed below 2.8·10⁻¹¹ bits.

A second check initialized Thin-PID and Tilde-PID at an optimal Gaussian coupling derived from the closed-form solution. Absolute differences in union information were at most 5.68·10⁻¹¹ bits across all trials, with mean absolute differences below 4.1·10⁻¹⁴ bits for both methods. These are agreement checks on population covariance matrices, not evidence about estimation from finite samples.

Runtime gains grow with source dimension

Runtime comparisons used Gaussian models with one target and two sources, increasing the dimension of one source while holding the target and the other source at 1000 dimensions, with all methods run on the same population covariance matrices and identical CPU resources. The paper reports speedups of up to three orders of magnitude, and its abstract states speedups exceeding two orders of magnitude for high-dimensional Gaussian BROJA PID.

The practical case in the paper is a real fMRI analysis: the main PID analysis took 14.5 minutes with Eigen-PID versus a projected 21 days with Thin-PID, and moving the computation from CPU to GPU reduced it further to 3.9 minutes. Because the closed form evaluates the exact population minimum, it also removes initialization, convergence, and stopping criteria from the procedure.

Positive definiteness and finite-sample bias remain

Applying Eigen-PID to data requires estimated source–target joint covariance matrices to satisfy the positive-definiteness assumption. The paper notes this fails for sample covariance estimates when N ≤ d_M + max(d_X, d_Y), where N is the sample size, and recommends regularization or dimensionality reduction to obtain positive-definite, well-conditioned estimates.

Even with well-conditioned covariances, replacing population covariances with sample covariances yields a plug-in estimator with finite-sample bias, as with existing Gaussian PID methods; the paper points to resampling-based corrections by Lorenz et al. (2026) and argues that the closed form makes those corrections practical at high dimension. For non-Gaussian data, the authors suggest Eigen-PID can replace Thin-PID inside the normalizing-flow pipeline of Zhao et al. (2025), removing the iterative step in the latent Gaussian space.

The eigenvalue split is the result to watch

The paper's distinctive claim is not merely that Gaussian BROJA PID can be computed faster, but that its structure is exposed: unique information is a sum of positive and negative parts of log₂ λ_i, so each target direction contributes to exactly one source's unique information depending on whether λ_i is above or below 1. Directions where the two posterior variances match contribute nothing unique to either source.

That makes the generalized eigenvalues of Γ_X and Γ_Y the quantity to inspect when deciding whether a Gaussian BROJA analysis is informative for a given dataset. The closed form is exact at the population level and requires positive-definite covariance estimates; within those conditions, it turns a costly optimization into an eigendecomposition whose per-direction output can be read directly.

Sources used in this answer