Constructing Unfaithful Coloured Gaussian DAG Models

Sammanfattning

Causal inference aims to incorporate cause and effect relationships when modelling different phenomena. This is represented by a directed acyclic graph (DAG) along with a linear equation representing the graph in terms of distributions. One often aims to learn the underlying DAG from a sample of the relevant distribution in the field of causal discovery. While older algorithms make strong assumptions about the model, newer state of the art algorithms aim to relax these assumptions. One prominent and restrictive assumption is known as faithfulness. Models without the faithfulness assumption are sparse in relation to all possible models, and no known method concerning the identification of such models for an arbitrary size exists. By imposing parameter restrictions in terms of graph colourings, we impose a method to generate unfaithful distributions. This is done by extending certain types of DAGs, where the parameter restriction imposes an unfaithful relationship. We also propose a conjecture regarding a complete classification of all unfaithful models. Using this method, we benchmark the causal discovery algorithms BOSS, GSP, and GRaSP on unfaithful data samples, and evaluate these algorithms based on True Positive Ratio (TPR), Structural Hamming Distance (SHD) and F1 score. The results show that these algorithms have similar performance on unfaithful relative faithful data sets for models with more variables, while models with a lower amount of variables exhibit larger differences. Our results indicate that the proposed method is a viable option when considering the performance of causal discovery algorithms when used on unfaithful data distributions, as well as a general framework to be used for understanding unfaithful models.