Causal Confirmation Measures: From Simpson's Paradox to COVID-19

Abstract

When we compare the influences of two causes on an outcome, if the conclusion from every group is against that from the conflation, we think there is Simpson's Paradox. The Existing Causal Inference Theory (ECIT) can make the overall conclusion consistent with the grouping conclusion by removing the confounder's influence to eliminate the paradox. The ECIT uses relative risk difference P_d = max(0, (R - 1)/R) (R denotes the risk ratio) as the probability of causation. In contrast, Philosopher Fitelson uses confirmation measure D (posterior probability minus prior probability) to measure the strength of causation. Fitelson concludes that from the perspective of Bayesian confirmation, we should directly accept the overall conclusion without considering the paradox. The author proposed a Bayesian confirmation measure b* similar to P_d before. To overcome the contradiction between the ECIT and Bayesian confirmation, the author uses the semantic information method with the minimum cross-entropy criterion to deduce causal confirmation measure Cc = (R - 1)/max(R, 1). Cc is like P_d but has normalizing property (between -1 and 1) and cause symmetry. It especially fits cases where a cause restrains an outcome, such as the COVID-19 vaccine controlling the infection. Some examples (about kidney stone treatments and COVID-19) reveal that P_d and Cc are more reasonable than D; Cc is more useful than P_d.

Keywords: Bayesian confirmation; COVID-19; Simpson’s Paradox; causal confirmation; causal inference; cross-entropy; risk measures; semantic information measure.

Figure 1

Illustrating Simpson’s Paradox. In each group, the success rate of x₂, P(y₁|x₂, g), is higher than that of x₁, P(y₁|x₁, g); however, using the method of finding the center of gravity, we can see that the overall success rate of x₂, P(y₁|x₂) = 0.65, is lower than that of x₁, P(y₁|x₁) = 0.7.

Figure 2

Three causal graphs: (a) for Example 2; (b) for Example 3; (c) for Examples 4 and 1.

Figure 3

Eliminating Simpson’s Paradox as the confounder exists by modifying the weighting coefficients. After replacing P(g_k|x_i) with P(g_k) (k = 1,2; i = 1,2), the overall conclusion is consistent with the grouping conclusion; the average success rate of x₂, P(y₁|do(x₂)) = 0.7, is higher than that of x₁, P(y₁|do(x₁)) = 0.65.

Figure 4

The truth function of s₁ includes a believable part with proportion b₁ and a disbelievable part with proportion b₁′.