Tree Dependent Identically Distributed Learning

We view a dataset of points or samples as having an underlying, yet unspecified, tree structure and exploit this assumption in learning problems. Such a tree structure assumption is equivalent to treating a dataset as being tree dependent identically distributed or tdid and preserves exchangeability. This extends traditional iid assumptions on data since each datum can be sampled sequentially after being conditioned on a parent. Instead of hypothesizing a single best tree structure, we infer a richer Bayesian posterior distribution over tree structures from a given dataset. We compute this posterior over (directed or undirected) trees via the Laplacian of conditional distributions between pairs of input data points. This posterior distribution is efficiently normalized by the Laplacian’s determinant and also facilitates novel maximum likelihood estimators, efficient expectations and other useful inference computations. In a classification setting, tdid assumptions yield a criterion that maximizes the determinant of a matrix of conditional distributions between pairs of input and output points. This leads to a novel classification algorithm we call the Maximum Determinant Machine. Unsupervised and supervised experiments are shown.