Keywords: counterfactual reasoning, inference complexity, twin networks, causal inference, (causal) treewidth, Bayesian networks
TL;DR: We show that counterfactual reasoning is no harder than associational/interventional reasoning by showing that the (causal) treewidth of a twin network is at most twice the (causal) treewidth of its base network plus one.
Abstract: A common form of counterfactual reasoning is based on the notion of twin network which is a causal graph that represents two worlds, one real and another imaginary. Information about the real world is used to update the joint distribution over the underlying causal mechanisms which is then used for hypothetical reasoning in the imaginary world. This is in contrast to associational and interventional reasoning which involve a causal graph over a single world that we shall call a base network. We study the complexity of counterfactual reasoning on twin networks in relation to the complexity of associational and interventional reasoning on base networks in the form of structural causal models (SCMs). We show that counterfactual reasoning is no harder than associational/interventional reasoning on fully specified SCMs in the context of two computational frameworks. One of these is based on the notion of treewidth and includes the classical variable elimination and jointree algorithms. The second, more recent framework is based on the notion of causal treewidth which is directed towards models that include SCMs. More specifically, we show that the (causal) treewidth of a twin network is at most twice the (causal) treewidth of its base network plus one. Hence, if associational/interventional reasoning is tractable on a fully specified SCM, then counterfactual reasoning is also tractable. We extend our results to counterfactual reasoning that requires contemplating more than two worlds and discuss applications of our results to counterfactual reasoning with partially specified SCMs (and data). We finally present empirical results that measure the gap between the complexities of counterfactual reasoning and associational/interventional reasoning on random SCMs.