D. Koutsoyiannis, C. Onof, Z. W. Kundzewicz, and A. Christofides, A stochastic approach to causality (Invited talk), AGU 2022 Fall Meeting, doi:10.13140/RG.2.2.25180.87681, American Geophysical Union, 2022.
We give a brief overview of conceptions of causality and attempts to find probabilistic characterizations of it. We argue that a useful criterion for causal links in open systems would apply to time-series of causally related phenomena, and that it only makes sense to seek necessary conditions for causality.
The criterion we develop uses an impulse response function g that relates two phenomena X and Y (for which contemporaneous time-series of observations are available) according to a convolution equation. The existence of such a function g which fulfils criteria of non-negativity and smoothness and leaves us with a residual random noise V with a minimal variance that is small compared to the variance of Y, is a condition for there being a causal or hen-or-egg link between two series (or two non-linear transforms thereof).
We understand a hen-or-egg case as one in which a clear causal link in one rather than the other direction cannot be identified. As seen in the figure, from the general case of a hen-or-egg causal system, we have as special cases potentially causal, potentially auticausal, and non-causal cases.
We demonstrate the plausibility of this necessary condition for causality by examining (i) a few artificial cases in which we show the role of the criteria we impose, (ii) the key hydrological causal link between rainfall and catchment runoff.
See also: https://agu.confex.com/agu/fm22/meetingapp.cgi/Paper/1081471