On Geometry of Information Flow for Causal Inference

Abstract

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 G e o C y → x . This avoids some of the boundedness issues that we show exist for the transfer entropy, T y → x . We will highlight our discussions with data developed from synthetic models of successively more complex nature: these include the Hénon map example, and finally a real physiological example relating breathing and heart rate function.

Transfer Entropy Based Causality from Head Motion to Eye Movement for Visual Scanning in Virtual Driving

“…Currently, for the two proposed causality measures, we consider more from the probabilistic perspective of prediction and complexity, based on the utilization of an information flow tool, transfer entropy, holding the idea of prediction by Wiener&Granger causality [27, 28]. Although it is not so clear whether Pearl’s causality driven by the idea of intervention [31] works well in the application context without intervention, as has been recognized in the literature [63], a combination of the ideas of information flow (transfer entropy) and Pearl’s intervention would be approached, constituting the authors’ future work.…”

“…Wiener&Granger causality [27,28]. Although it is not so clear whether Pearl's causality driven by the idea of intervention [31] works well in the application context without intervention, as has been recognized in the literature [63], a combination of the ideas of information flow (transfer entropy) and Pearl's intervention would be approached, constituting the authors' future work.…”

The Role of Instrumental Variables in Causal Inference Based on Independence of Cause and Mechanism

“…There exist various applications of causal inference. Thus, [24] provides a geometric interpretation of information flow as a causal inference. Speaking of probabilistic causal inference approaches, we would like to mention [25], which is a survey considering probabilistic causal dependencies among variables.…”