Norbert Wiener's legacy in the study and inference of causation

Norbert Wiener has pioneered the study of causality and his ideas are still shaping this active area of research. The present article explores the concept of 'inferred causation' and demonstrates the continued impact of Wiener's work. It shows how the statistical notion of causation introduced by Clive Granger in the 70's and the counterfactual notion of causation introduced by Judea Pearl in the 90's can be interpreted and merged under a single unifying framework that is based on Wiener filtering. Determining whether a given process 'causes' another has been a central subject of philosophical debate since even before the time of David Hume and Immanuel Kant. Aside from purely philosophical or gnoseological considerations, practicing scientists are content to be able to infer causation in operative ways. In practical terms, if it is possible to isolate the processes of interest within a controlled laboratory environment, causality can be inferred by performing a sufficient number of experiments. However, a deeper understanding of the meaning of causation is required if the processes can not be perfectly isolated, not all variables can be measured or only passive observations are available. Traditionally, Clive Granger is credited with the introduction of the first quantitative statistical tests to infer causation from passive observations. In Granger's formulation, one variable is causal to another, if the ability to predict the second variable is significantly improved by incorporating information about the first. As Granger himself acknowledges in his seminal paper, both the mathematical formalization and the fundamental ideas of his notion of causality have been deeply inspired by the contributions of Wiener in the area of estimation and prediction. More recently, a different approach to the study of causality has been developed by the computer scientist and philosopher Judea Pearl. Central to Pearl's work is the description of probability distributions and conditional independence among random variables by using directed acyclic graphs. Over this class of graphs, Pearl has introduced a semantic language and computational rules to formally describe the concept of causality in a network of partially observed variables. These computational rules are usually known as 'do-calculus.' Typically, Pearl's approach is static, in the sense that there is no time variable defined in the probabilistic model, and is not well-suited to consider feedback loops. Granger's approach, instead, is based on one-step ahead predictors exploiting the fact that the processes are dynamical systems, but is not capable of dealing with unobserved quantities. In this article, we provide a combination of both approaches. We first show how, by using certain variations of the Wiener filter, specific sparsity properties can be represented over directed graphs where loops can be present as well. These results are in line with the results obtained by Pearl for his graphical models. Under relatively mild hypotheses, we, as well, provide an interpretation for a generalized (multivariate) notion of Granger causality over these graphs. Finally, we prove that these Wiener filters can be used within the framework provided by do-calculus and seamlessly take into account scenarios where feedback loops are also present.