Computing Lower and Upper Bounds on the Probability of Causal Statements

Causal discovery provides an opportunity to infer causal relationships from purely observational data and to predict the effect of interventions. Constraint-based methods for causal discovery exploit conditional (in)dependencies to infer the direction of causal relationships. They typically work through forward chaining: given some causal statements, others can be inferred by applying relatively straightforward causal logic such as transitivity and acyclicity. Starting from the premise that we can estimate reliabilities for base causal statements, we propose a novel approach to estimate the reliability of novel statements inferred by forward chaining. Since reliabilities for base statements are clearly dependent, if only because inferred from the same data, exact computation is infeasible. However, lending ideas from the area of imprecise probability theory, we can compute bounds on the reliabilities on inferred statements. Specifically, we make use of the good old Fréchet inequalities and discuss two different variants: greedy and delayed. In simulation experiments, we show that the delayed variant, at the expense of more bookkeeping and computation time, does provide slightly tighter intervals. We illustrate our method on a real-world data set about attention deficit/hyperactivity disorder.