Evaluating causes of effects by posterior effects of causes

For the case with a single causal variable, Dawid et al. (2014) defined the probability of causation, and Pearl (2000) defined the probability of necessity to assess the causes of effects. For a case with multiple causes that could affect each other, this paper defines the posterior total and direct causal effects based on the evidence observed for post-treatment variables, which could be viewed as measurements of causes of effects. Posterior causal effects involve the probabilities of counterfactual variables. Thus, as with the probability of causation, the probability of necessity and direct causal effects, the identifiability of posterior total and direct causal effects requires more assumptions than the identifiability of traditional causal effects conditional on pre-treatment variables. We present assumptions required for the identifiability of posterior causal effects and provide identification equations. Further, when the causal relationships between multiple causes and an endpoint can be depicted by causal networks, we can simplify both the required assumptions and the identification equations of the posterior total and direct causal effects. Finally, using numerical examples, we compare the posterior total and direct causal effects with other measures for evaluating the causes of effects and the population attributable risks.