{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,7]],"date-time":"2026-03-07T14:17:28Z","timestamp":1772893048742,"version":"3.50.1"},"reference-count":16,"publisher":"MDPI AG","issue":"3","license":[{"start":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T00:00:00Z","timestamp":1645660800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP17K06446"],"award-info":[{"award-number":["JP17K06446"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP19K04914"],"award-info":[{"award-number":["JP19K04914"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["JP19K14989"],"award-info":[{"award-number":["JP19K14989"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>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.<\/jats:p>","DOI":"10.3390\/e24030328","type":"journal-article","created":{"date-parts":[[2022,2,24]],"date-time":"2022-02-24T21:11:07Z","timestamp":1645737067000},"page":"328","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":12,"title":["Probability Distribution on Full Rooted Trees"],"prefix":"10.3390","volume":"24","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-0553-7910","authenticated-orcid":false,"given":"Yuta","family":"Nakahara","sequence":"first","affiliation":[{"name":"Center for Data Science, Waseda University, Shinjuku-ku 169-8050, Tokyo, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-2256-4598","authenticated-orcid":false,"given":"Shota","family":"Saito","sequence":"additional","affiliation":[{"name":"Faculty of Informatics, Gunma University, Maebashi-shi 371-8510, Gunma, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Akira","family":"Kamatsuka","sequence":"additional","affiliation":[{"name":"Department of Information Science, Shonan Institute of Technology, Fujisawa-shi 251-8511, Kanagawa, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Toshiyasu","family":"Matsushima","sequence":"additional","affiliation":[{"name":"Department of Applied Mathematics, Waseda University, Shinjuku-ku 169-8555, Tokyo, Japan"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,2,24]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"653","DOI":"10.1109\/18.382012","article-title":"The context-tree weighting method: Basic properties","volume":"41","author":"Willems","year":"1995","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1649","DOI":"10.1109\/TCSVT.2012.2221191","article-title":"Overview of the High Efficiency Video Coding (HEVC) Standard","volume":"22","author":"Sullivan","year":"2012","journal-title":"IEEE Trans. Circuits Syst. Video Technol."},{"key":"ref_3","unstructured":"Breiman, L., Friedman, J., Stone, C.J., and Olshen, R.A. (1984). Classification and Regression Trees, CRC Press."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Chen, T., and Guestrin, C. (2016, January 13\u201317). XGBoost: A Scalable Tree Boosting System. Proceedings of the 22nd ACM SIGKDD International Conference on Knowledge Discovery and Data Mining, San Francisco, CA, USA.","DOI":"10.1145\/2939672.2939785"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5","DOI":"10.1023\/A:1010933404324","article-title":"Random Forests","volume":"45","author":"Breiman","year":"2001","journal-title":"Mach. 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Proceedings of the 2009 IEEE International Symposium on Information Theory, Seoul, Korea.","DOI":"10.1109\/ISIT.2009.5205677"},{"key":"ref_10","doi-asserted-by":"crossref","unstructured":"Papageorgiou, I., Kontoyiannis, I., Mertzanis, L., Panotopoulou, A., and Skoularidou, M. (2021, January 12\u201320). Revisiting Context-Tree Weighting for Bayesian Inference. Proceedings of the 2021 IEEE International Symposium on Information Theory (ISIT), Melbourne, Australia.","DOI":"10.1109\/ISIT45174.2021.9518189"},{"key":"ref_11","unstructured":"Kontoyiannis, I., Mertzanis, L., Panotopoulou, A., Papageorgiou, I., and Skoularidou, M. (2020). Bayesian Context Trees: Modelling and exact inference for discrete time series. arXiv."},{"key":"ref_12","doi-asserted-by":"crossref","unstructured":"Nakahara, Y., and Matsushima, T. (2021). A Stochastic Model for Block Segmentation of Images Based on the Quadtree and the Bayes Code for It. Entropy, 23.","DOI":"10.3390\/e23080991"},{"key":"ref_13","doi-asserted-by":"crossref","unstructured":"Dobashi, N., Saito, S., Nakahara, Y., and Matsushima, T. (2021). Meta-Tree Random Forest: Probabilistic Data-Generative Model and Bayes Optimal Prediction. Entropy, 23.","DOI":"10.3390\/e23060768"},{"key":"ref_14","unstructured":"Kenneth, R. (2011). Discrete Mathematics and Its Applications, McGraw-Hill Science. [7th ed.]."},{"key":"ref_15","unstructured":"Bishop, C. (2006). Pattern Recognition and Machine Learning, Springer."},{"key":"ref_16","doi-asserted-by":"crossref","unstructured":"Cover, T.M., and Thomas, J.A. (2006). 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