When causation does not imply correlation: robust violations of the faithfulness axiom

Post date: Mar 8, 2012 9:44:38 PM

Summary of the presentation of Richard Kennaway:

Current methods of inferring causal information from correlational data assume that causation implies correlation: that whenever there is a causal connection between two variables, their correlation must be non-zero. More precisely, it is claimed that a zero correlation in the presence of causal influences can only arise by the unlikely chance (a chance with probability zero) of multiple causal connections between the two variables exactly cancelling out. This is the faithfulness axiom.

We exhibit two counterexamples to this axiom: classes of systems in which faithfulness is robustly violated. These systems exhibit correlations indistinguishable from zero between variables that are strongly causally connected, and very high correlations between variables that have no direct causal connection, only a connection via causal links between uncorrelated variables. Furthermore, these systems are not of any artificially contrived sort. On the contrary, the equations defining them and real physical systems exemplifying them are commonplace.

The first example is that of a bounded differentiable variable and its first derivative, or a discrete time series and its first difference. The second example is control systems. In control systems there is a systematic tendency to produce low or zero correlations between variables that are physically directly connected, together with very high correlations between variables whose only causal connections are indirect, proceeding via those low-correlation links. That this is even possible may sound paradoxical, but it is inherent in the way that these systems operate, and readily demonstrated by mathematical analysis, numerical simulation, and physical measurement.

All of these counterexamples violate one of more of the preconditions required for various published methods of causal inference to be applied. There is thus no contradiction of those results, but a proof of a limitation of their scope.