Learning Staged Trees from Incomplete Data

Staged trees are probabilistic graphical models capable of representing any class of non-symmetric independence via a coloring of its vertices. Several structural learning routines have been defined and implemented to learn staged trees from data, under the frequentist or Bayesian paradigm. They assume a data set has been observed fully and, in practice, observations with missing entries are either dropped or imputed before learning the model. Here, we introduce the first algorithms for staged trees that handle missingness within the learning of the model. To this end, we characterize the likelihood of staged tree models in the presence of missing data and discuss pseudo-likelihoods that approximate it. A structural expectation-maximization algorithm estimating the model directly from the full likelihood is also implemented and evaluated. A computational experiment showcases the performance of the novel learning algorithms, demonstrating that it is feasible to account for different missingness patterns when learning staged trees.

Positive dependence in qualitative probabilistic networks

Qualitative probabilistic networks (QPNs) combine the conditional independence assumptions of Bayesian networks with the `qualitative' properties of positive and negative dependence. They attempt to formalise various intuitive properties of positive dependence to allow inferences over a large network of variables. However, we highlight a key mistake in the QPN literature which means that most inferences made by a QPN are not mathematically true. We also discuss how to redefine a QPN in order to fix this issue.