Information Imbalance Gain (IIG)

Wenbo Lyu

Last update: 2026-08-15
Last run: 2026-08-15

Introduction

The Information Imbalance Gain (IIG) quantifies the information that a variable \(X\) provides about the future state of another variable \(Y\). To test the causation \(X \rightarrow Y\), IIG compares the neighborhood structure of the future state of \(Y\) with that of an augmented present state containing both \(X\) and \(Y\).

The basic quantity is the Information Imbalance between two distance spaces \(A\) and \(B\):

\[ \Delta(A \rightarrow B) = \frac{2}{N} \left\langle r^B \mid r^A \leq k \right\rangle, \]

where \(r^A\) and \(r^B\) are the distance ranks in spaces \(A\) and \(B\), respectively, and \(k\) is the number of nearest neighbors considered. A small \(\Delta(A\rightarrow B)\) indicates that points close in \(A\) also tend to be close in \(B\). Thus, the distance rank in one space provides information about the structure of another space.

For \(X \rightarrow Y\), let \(d_Y(0)\) denote the distance space constructed from the present state of \(Y\), and let \(d_{XY}^{\alpha}(0)\) denote the corresponding space after incorporating \(X\) with a relative scaling parameter \(\alpha\). The future state of \(Y\) is represented by \(d_Y(\tau)\), where \(\tau\) is the prediction horizon. IIG therefore evaluates

\[ \Delta(\alpha) = \Delta \left( d_{XY}^{\alpha}(0) \rightarrow d_Y(\tau) \right). \]

If \(X\) contains information about the future of \(Y\), adding \(X\) should reduce the Information Imbalance. The resulting Imbalance Gain is

\[ \mathrm{IIG}(X\rightarrow Y) = \frac{ \Delta(0)-\min_{\alpha}\Delta(\alpha) }{ \Delta(0) }. \]

Here, \(\Delta(0)\) represents prediction using \(Y\) alone, whereas \(\min_{\alpha}\Delta(\alpha)\) represents the best prediction obtained after incorporating information from \(X\). Consequently, \(\mathrm{IIG}(X\rightarrow Y)=0\) indicates no gain from adding \(X\), while a positive value indicates that \(X\) provides additional information about the future state of \(Y\).

The same calculation can be performed in the opposite direction, \(Y\rightarrow X\), allowing directional information transfer to be assessed for both directions.

Example Cases