Semi-Supervised Learning, Causality and the Conditional Cluster Assumption

While the success of semi-supervised learning (SSL) is still not fully understood, Sch\"olkopf et al. (2012) have established a link to the principle of independent causal mechanisms. They conclude that SSL should be impossible when predicting a target variable from its causes, but possible when predicting it from its effects. Since both these cases are somewhat restrictive, we extend their work by considering classification using cause and effect features at the same time, such as predicting a disease from both risk factors and symptoms. While standard SSL exploits information contained in the marginal distribution of the inputs (to improve our estimate of the conditional distribution of target given inputs), we argue that in our more general setting we can use information in the conditional of effect features given causal features. We explore how this insight generalizes the previous understanding, and how it relates to and can be exploited for SSL.