# Probability Distribution on Rooted Trees

arXiv: https://arxiv.org/abs/2201.09460 · HTML: https://mlsanity.com/paper/2201.09460/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 3 of its 15 references that are in the map.
- First posted on arXiv 24 Jan 2022.

## Builds on (most cited first, 3 of 3)

- [XGBoost: A Scalable Tree Boosting System](https://mlsanity.com/paper/1603.02754/) (2016)
- [Bayesian Context Trees: Modelling and exact inference for discrete time series](https://mlsanity.com/paper/2007.14900/) (2020)
- [Probability Distribution on Full Rooted Trees](https://mlsanity.com/paper/2109.12825/) (2021)

## Abstract (arXiv)

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.