  Recent work has discussed the limitations of counterfactual explanations to
recommend actions for algorithmic recourse, and argued for the need of taking
causal relationships between features into consideration. Unfortunately, in
practice, the true underlying structural causal model is generally unknown. In
this work, we first show that it is impossible to guarantee recourse without
access to the true structural equations. To address this limitation, we propose
two probabilistic approaches to select optimal actions that achieve recourse
with high probability given limited causal knowledge (e.g., only the causal
graph). The first captures uncertainty over structural equations under additive
Gaussian noise, and uses Bayesian model averaging to estimate the
counterfactual distribution. The second removes any assumptions on the
structural equations by instead computing the average effect of recourse
actions on individuals similar to the person who seeks recourse, leading to a
novel subpopulation-based interventional notion of recourse. We then derive a
gradient-based procedure for selecting optimal recourse actions, and
empirically show that the proposed approaches lead to more reliable
recommendations under imperfect causal knowledge than non-probabilistic
baselines.
