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Probability Distribution on Full Rooted Trees.

Yuta Nakahara1, Shota Saito2, Akira Kamatsuka3

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Summary

This study introduces a new probability distribution for full rooted trees, addressing model selection challenges in machine learning and data compression. The proposed method enables optimal model selection using Bayes decision theory and recursive calculations.

Keywords:
Bayes decision theoryBayes statisticsrecursive algorithmrooted trees

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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.