  Knowledge of the underlying causal relations is essential for inferring the
effect of interventions in complex systems. In a widely studied approach,
structural causal models postulate noisy functional relations among interacting
variables, where the underlying causal structure is then naturally represented
by a directed graph whose edges indicate direct causal dependencies. In the
typical application, this underlying causal structure must be learned from
data, and thus, the remaining structure uncertainty needs to be incorporated
into causal inference in order to draw reliable conclusions. In recent work,
test inversions provide an ansatz to account for this data-driven model choice
and, therefore, combine structure learning with causal inference. In this
article, we propose the use of dual likelihood to greatly simplify the
treatment of the involved testing problem. Indeed, dual likelihood leads to a
closed-form solution for constructing confidence regions for total causal
effects that rigorously capture both sources of uncertainty: causal structure
and numerical size of nonzero effects. The proposed confidence regions can be
computed with a bottom-up procedure starting from sink nodes. To render the
causal structure identifiable, we develop our ideas in the context of linear
causal relations with equal error variances.
