Probability Distribution on Full Rooted Trees

TL;DR

This paper introduces a new probability distribution on full rooted trees, enabling Bayesian model selection and averaging in hierarchical models across various applications.

Contribution

It generalizes existing distributions with a parametric form suitable for recursive calculations of properties like expectation and posterior, applicable to diverse fields.

Findings

Derived a new generalized distribution for full rooted trees.

Developed recursive methods for calculating distribution properties.

Facilitated Bayesian model selection and averaging using the distribution.

Abstract

The recursive and hierarchical structure of full rooted trees is applicable to represent statistical models in various areas, such as data compression, image processing, and machine learning. In most of these cases, the full rooted tree is not a random variable; as such, model selection to avoid overfitting becomes problematic. A method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on the Bayes decision theory. For example, by assigning a low prior probability to a complex model, the maximum a posteriori estimator prevents the selection of the complex one. Furthermore, we can average all the models weighted by their posteriors. In this paper, we propose a probability distribution on a set of full rooted trees. Its parametric representation is suitable for calculating the properties of our distribution using recursive functions, such as the mode, expectation, and posterior distribution. Although such distributions have been proposed in previous studies, they are only applicable to specific applications. Therefore, we extract their mathematically essential components and derive new generalized methods to calculate the expectation, posterior distribution, etc.