Probability Distribution on Rooted Trees

TL;DR

This paper introduces a generalized probability distribution for rooted trees with fixed maximum children and depth, along with recursive methods to evaluate their properties, enhancing hierarchical modeling in machine learning.

Contribution

It proposes a novel probability distribution for rooted trees with constraints on children and depth, and derives recursive evaluation methods, extending prior full-tree distributions.

Findings

Provides a recursive evaluation method for the new distribution.

Enables modeling of hierarchical structures with fixed constraints.

Improves over previous full-tree probability models.

Abstract

The hierarchical and recursive expressive capability of rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. On the other hand, such hierarchical expressive capability causes a problem in tree selection to avoid overfitting. One unified approach to solve this is a Bayesian approach, on which the rooted tree is regarded as a random variable and a direct loss function can be assumed on the selected model or the predicted value for a new data point. However, all the previous studies on this approach are based on the probability distribution on full trees, to the best of our knowledge. In this paper, we propose a generalized probability distribution for any rooted trees in which only the maximum number of child nodes and the maximum depth are fixed. Furthermore, we derive recursive methods to evaluate the characteristics of the probability distribution without any approximations.