Comments on: "Hybrid Semiparametric Bayesian Networks"

This is an interesting paper that distils structure learning in Bayesian networks (BNs) and kernel methods in a quest to produce more flexible distributional assumptions. Conditional (linear) Gaussian Bayesian networks (CGBNs) have been well explored in the literature for some time, to the point that they now appear in many recent textbooks [1-3]. The authors address one of the key limitations of CGBNS, that they can only capture linear dependencies between the continuous variables they contain, and remove it by replacing (mixtures of) linear regression models with more general kernel densities. Dependencies between discrete variables were already flexible, because the conditional probability tables that parametrise them essentially act as a saturated model[4]. It is not obvious that more flexibility will produce better models for whatever task we have in mind: it can also lead to overfitting, instability and hyperparameter tuning problems.