Adaptive Bayesian Reticulum

Neural Networks and Random Forests: two popular techniques for supervised learning that are seemingly disconnected in their formulation and optimization method, have recently been linked in a single construct. The connection pivots on assembling an artificial Neural Network with nodes that allow for a gate-like function to mimic a tree split, optimized using the standard approach of recursively applying the chain rule to update its parameters. Yet two main challenges have impeded wide use of this hybrid approach: \emph{(a)} the inability of global gradient descent techniques to optimize hierarchical parameters (as introduced by the gate function); and \emph{(b)} the construction of the tree structure, which has relied on standard decision tree algorithms to learn the network topology or incrementally (and heuristically) searching the space at random. We propose a probabilistic construct that exploits the idea of a node's \emph{unexplained potential} (the total error channeled through the node) in order to decide where to expand further, mimicking the standard tree construction in a Neural Network setting, alongside a modified gradient descent that first locally optimizes an expanded node before a global optimization. The probabilistic approach allows us to evaluate each new split as a ratio of likelihoods that balance the statistical improvement in explaining the evidence against the additional model complexity --- thus providing a natural stopping condition. The result is a novel classification and regression technique that leverages the strength of both: a tree-structure that grows naturally and is simple to interpret with the plasticity of Neural Networks that allow for soft margins and slanted boundaries.

A Bayesian Decision Tree Algorithm

Bayesian Decision Trees are known for their probabilistic interpretability. However, their construction can sometimes be costly. In this article we present a general Bayesian Decision Tree algorithm applicable to both regression and classification problems. The algorithm does not apply Markov Chain Monte Carlo and does not require a pruning step. While it is possible to construct a weighted probability tree space we find that one particular tree, the greedy-modal tree (GMT), explains most of the information contained in the numerical examples. This approach seems to perform similarly to Random Forests.