We study the identification of causal effects, motivated by two improvements to identifiability which can be attained if one knows that some variables in a causal graph are functionally determined by their parents (without needing to know the specific functions). First, an unidentifiable causal effect may become identifiable when certain variables are functional. Second, certain functional variables can be excluded from being observed without affecting the identifiability of a causal effect, which may significantly reduce the number of needed variables in observational data. Our results are largely based on an elimination procedure which removes functional variables from a causal graph while preserving key properties in the resulting causal graph, including the identifiability of causal effects.

We discuss some recent advances on combining data and knowledge in the context of supervised learning using Bayesian networks. A first set of advances concern the computational efficiency of learning and inference, and they include a software-level boost based on compiling Bayesian network structures into tractable circuits in the form of tensor graphs, and algorithmic improvements based on exploiting a type of knowledge called unknown functional dependencies. The used tensor graphs capitalize on a highly optimized tensor operation (matrix multiplication) which brings orders of magnitude speedups in circuit training and evaluation. The exploitation of unknown functional dependencies yields exponential reductions in the size of tractable circuits and gives rise to the notion of causal treewidth for offering a corresponding complexity bound. Beyond computational efficiency, we discuss empirical evidence showing the promise of learning from a combination of data and knowledge, in terms of data hungriness and robustness against noise perturbations. Sometimes, however, an accurate Bayesian network structure may not be available due to the incompleteness of human knowledge, leading to modeling errors in the form of missing dependencies or missing variable values. On this front, we discuss another set of advances for recovering from certain types of modeling errors. This is achieved using Testing Bayesian networks which dynamically select parameters based on the input evidence, and come with theoretical guarantees on full recovery under certain conditions.

We study the computational complexity of counterfactual reasoning in relation to the complexity of associational and interventional reasoning on structural causal models (SCMs). We show that counterfactual reasoning is no harder than associational or interventional reasoning on fully specified SCMs in the context of two computational frameworks. The first framework is based on the notion of treewidth and includes the classical variable elimination and jointree algorithms. The second framework is based on the more recent and refined notion of causal treewidth which is directed towards models with functional dependencies such as SCMs. Our results are constructive and based on bounding the (causal) treewidth of twin networks---used in standard counterfactual reasoning that contemplates two worlds, real and imaginary---to the (causal) treewidth of the underlying SCM structure. In particular, we show that the latter (causal) treewidth is no more than twice the former plus one. Hence, if associational or interventional reasoning is tractable on a fully specified SCM then counterfactual reasoning is tractable too. We extend our results to general counterfactual reasoning that requires contemplating more than two worlds and discuss applications of our results to counterfactual reasoning with partially specified SCMs that are coupled with data. We finally present empirical results that measure the gap between the complexities of counterfactual reasoning and associational/interventional reasoning on random SCMs.

Causal treewidth is a recently introduced notion allowing one to speed up Bayesian network inference and to bound its complexity in the presence of functional dependencies (causal mechanisms) whose identities are unknown. Causal treewidth is no greater than treewidth and can be bounded even when treewidth is unbounded. The utility of causal treewidth has been illustrated recently in the context of causal inference and model-based supervised learning. However, the current definition of causal treewidth is descriptive rather than perspective, therefore limiting its full exploitation in a practical setting. We provide an extensive study of causal treewidth in this paper which moves us closer to realizing the full computational potential of this notion both theoretically and practically.