# On Geometry of Information Flow for Causal Inference

arXiv: https://arxiv.org/abs/2002.02078 · HTML: https://mlsanity.com/paper/2002.02078/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Feb 2020.

## Built on by (most cited first, 4 of 4)

- [On Learning what to Learn: heterogeneous observations of dynamics and establishing (possibly causal) relations among them](https://mlsanity.com/paper/2406.06812/) (2024)
- [Swarm Analytics: Designing Information Markers to Characterise Swarm Systems in Shepherding Contexts](https://mlsanity.com/paper/2208.12386/) (2022)
- [Reconciling Causality and Non-Equilibrium Thermodynamics with Hamiltonian Causal Models](https://mlsanity.com/paper/2606.04822/) (2026)
- [Interventional Score Geometry for Causal Inference](https://mlsanity.com/paper/2607.21914/) (2026)

## Abstract (arXiv)

Causal inference is perhaps one of the most fundamental concepts in science, beginning originally from the works of some of the ancient philosophers, through today, but also weaved strongly in current work from statisticians, machine learning experts, and scientists from many other fields. This paper takes the perspective of information flow, which includes the Nobel prize winning work on Granger-causality, and the recently highly popular transfer entropy, these being probabilistic in nature. Our main contribution will be to develop analysis tools that will allow a geometric interpretation of information flow as a causal inference indicated by positive transfer entropy. We will describe the effective dimensionality of an underlying manifold as projected into the outcome space that summarizes information flow. Therefore contrasting the probabilistic and geometric perspectives, we will introduce a new measure of causal inference based on the fractal correlation dimension conditionally applied to competing explanations of future forecasts, which we will write GeoCyrightarrow x. This avoids some of the boundedness issues that we show exist for the transfer entropy, Tyrightarrow x. We will highlight our discussions with data developed from synthetic models of successively more complex nature: then include the H'enon map example, and finally a real physiological example relating breathing and heart rate function. Keywords: Causal Inference; Transfer Entropy; Differential Entropy; Correlation Dimension; Pinsker's Inequality; Frobenius-Perron operator.