A note on unrestricted Chow–Liu tree selection for mixed discrete–continuous data
摘要
The Chow–Liu algorithm summarises the dependency structure of mixed discrete–continuous data as a tree of the strongest pairwise associations. A structural restriction excludes trees in which a continuous variable connects two discrete variables, because such a path complicates the later density-fitting step. We show that this restriction is unnecessary for tree selection, which depends only on pairwise mutual-information values, well-defined for every combination of variable types. The restriction applies only to the later density-fitting step. We prove consistency of the unrestricted procedure and give finite-sample error bounds under a tail assumption on the pairwise mutual-information estimates. On synthetic mixed graphs and a synthetic gene-network benchmark, the unrestricted procedure recovers trees that the restricted procedure cannot. On the fully discrete Asia Bayesian benchmark, where the restriction is inactive by construction, the pipeline attains the tree-approximation ceiling, the best any spanning tree can achieve. A breast-cancer gene-expression dataset is included as an illustration, since no ground truth is available. The comparison with a discretisation-based estimator is regime-dependent. In the sample-size sweep, the discretisation estimator performs better at