TY - JOUR

T1 - Probability Distribution on Full Rooted Trees

AU - Nakahara, Yuta

AU - Saito, Shota

AU - Kamatsuka, Akira

AU - Matsushima, Toshiyasu

N1 - Funding Information:
Funding: This research was funded by JSPS KAKENHI, grant numbers JP17K06446, JP19K04914, and JP19K14989.
Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/3

Y1 - 2022/3

N2 - The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, 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 is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on 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.

AB - The recursive and hierarchical structure of full rooted trees is applicable to statistical models in various fields, 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 is problematic. One method to solve this problem is to assume a prior distribution on the full rooted trees. This enables the optimal model selection based on 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.

KW - Bayes decision theory

KW - Bayes statistics

KW - Recursive algorithm

KW - Rooted trees

UR - http://www.scopus.com/inward/record.url?scp=85125323804&partnerID=8YFLogxK

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U2 - 10.3390/e24030328

DO - 10.3390/e24030328

M3 - Article

AN - SCOPUS:85125323804

SN - 1099-4300

VL - 24

JO - Entropy

JF - Entropy

IS - 3

M1 - 328

ER -