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Counterfactual inference is a powerful tool, capable of solving challengingproblems in high-profile sectors. To perform counterfactual inference, onerequires knowledge of the underlying causal mechanisms. However, causalmechanisms cannot be uniquely determined from observations and interventionsalone. This raises the question of how to choose the causal mechanisms so thatresulting counterfactual inference is trustworthy in a given domain. Thisquestion has been addressed in causal models with binary variables, but thecase of categorical variables remains unanswered. We address this challenge byintroducing for causal models with categorical variables the notion ofcounterfactual ordering, a principle that posits desirable properties causalmechanisms should posses, and prove that it is equivalent to specificfunctional constraints on the causal mechanisms. To learn causal mechanismssatisfying these constraints, and perform counterfactual inference with them,we introduce deep twin networks. These are deep neural networks that, whentrained, are capable of twin network counterfactual inference -- an alternativeto the abduction, action, & prediction method. We empirically test our approachon diverse real-world and semi-synthetic data from medicine, epidemiology, andfinance, reporting accurate estimation of counterfactual probabilities whiledemonstrating the issues that arise with counterfactual reasoning whencounterfactual ordering is not enforced.

Estimating Categorical Counterfactuals via Deep Twin Networks