Confidence in causal inference under structure uncertainty in linear causal models with equal variances

Inferring the effect of interventions within complex systems is a fundamental problem of statistics. A widely studied approach uses structural causal models that postulate noisy functional relations among a set of interacting variables. The underlying causal structure is then naturally represented by a directed graph whose edges indicate direct causal dependencies. In a recent line of work, additional assumptions on the causal models have been shown to render this causal graph identifiable from observational data alone. One example is the assumption of linear causal relations with equal error variances that we will take up in this work. When the graph structure is known, classical methods may be used for calculating estimates and confidence intervals for causal-effects. However, in many applications, expert knowledge that provides an a priori valid causal structure is not available. Lacking alternatives, a commonly used two-step approach first learns a graph and then treats the graph as known in inference. This, however, yields confidence intervals that are overly optimistic and fail to account for the data-driven model choice. We argue that to draw reliable conclusions, it is necessary to incorporate the remaining uncertainty about the underlying causal structure in confidence statements about causal-effects. To address this issue, we present a framework based on test inversion that allows us to give confidence regions for total causal-effects that capture both sources of uncertainty: causal structure and numerical size of non-zero effects.