How to Measure Causal Influence in Gaussian Data?

Directed information is a tool to detect who influences whom in two time series. When the data are real numbers, like sensor readings or audio signals, estimating these values with few samples is tricky. A recent paper tackles this for sequences of Gaussian vectors—sets of numbers following a bell‑shaped distribution. The authors derived a clear formula for the causally conditioned directed information rate, based on optimal prediction: forecasting the next value using all past data. From this formula they built a practical estimator. The striking result is that the estimator’s error shrinks as N⁻¹ᐟ²·log(N), where N is the total number of observations. In plain terms, doubling the data cuts the error by about 30 %, with a small extra due to the logarithm. This is a “non‑asymptotic” bound, meaning it holds even for moderate sample sizes, not only at infinity. The finding paves the way for more reliable causality analysis in fields like neuroscience, finance, or climate science, where data are often Gaussian and series are finite.

The causally conditioned directed information (CDI) rate for Gaussian processes can be expressed via the conditional covariance matrix of prediction residuals. Practically, one builds the linear minimum mean‑square error (LMMSE) predictor that minimizes the quadratic error between the future value and its estimate based on the past. The prediction error matrix Σ_e is given by Σ_X|Y = Σ_X - Σ_XY Σ_Y⁻¹ Σ_YX, where Σ_X and Σ_Y are the covariance matrices of vectors X and Y, and Σ_XY their cross‑covariance. The CDI rate then equals ½ log det(Σ_X|Y)⁻¹ Σ_X|Y, which matches the differential entropy difference between the unconditioned and conditioned processes. The proposed estimator computes these matrices from finite samples, using unbiased covariance estimates and regularization to ensure invertibility. Concentration analysis of covariance matrices shows that, with high probability, the deviation between the sample estimate and the true matrix is O(√(log N / N)). Plugging this deviation into the CDI formula yields an overall error of O(N⁻¹ᐟ² log N). This result leverages random matrix theory and matrix Bernstein inequalities, extending classic Cramér‑Rao bounds to non‑asymptotic settings. Consequently, even with a few hundred samples, the estimator provides a reliable causal influence measure, provided Gaussianity and stationarity assumptions hold empirically.