Probability Distribution on Full Rooted Trees.

Area of Science:

• Statistics
• Computer Science
• Machine Learning

Background:

• Full rooted trees have recursive and hierarchical structures applicable to various statistical models.
• Model selection for non-random trees is problematic due to overfitting.
• Prior distributions on trees can enable optimal model selection via Bayes decision theory.

Purpose of the Study:

• To propose a novel probability distribution on the set of full rooted trees.
• To develop generalized methods for calculating distribution properties like expectation and posterior distribution.
• To address limitations of existing distributions applicable only to specific domains.

Main Methods:

• Developing a parametric representation for the probability distribution.
• Utilizing recursive functions for calculating distribution properties.
• Extracting essential mathematical components from prior distributions for generalization.

Main Results:

• A new probability distribution on full rooted trees is proposed.
• Parametric representation allows efficient calculation of mode, expectation, and posterior distribution.
• Generalized methods are derived for calculating key statistical properties.

Conclusions:

• The proposed probability distribution and calculation methods offer a generalized approach to tree model selection.
• This framework enhances statistical modeling in fields utilizing hierarchical structures.
• The approach facilitates more robust and accurate model selection, mitigating overfitting.