{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:24:15Z","timestamp":1760243055633,"version":"build-2065373602"},"reference-count":13,"publisher":"MDPI AG","issue":"6","license":[{"start":{"date-parts":[[2015,6,10]],"date-time":"2015-06-10T00:00:00Z","timestamp":1433894400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>Based on geometric invariance properties, we derive an explicit prior distribution for the parameters of multivariate linear regression problems in the absence of further prior information. The problem is formulated as a rotationally-invariant distribution of \\(L\\)-dimensional hyperplanes in \\(N\\) dimensions, and the associated system of partial differential equations is solved. The derived prior distribution generalizes the already known special cases, e.g., 2D plane in three dimensions.<\/jats:p>","DOI":"10.3390\/e17063898","type":"journal-article","created":{"date-parts":[[2015,6,10]],"date-time":"2015-06-10T14:05:21Z","timestamp":1433945121000},"page":"3898-3912","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["General Hyperplane Prior Distributions Based on Geometric Invariances for Bayesian Multivariate Linear Regression"],"prefix":"10.3390","volume":"17","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-8867-1014","authenticated-orcid":false,"given":"Udo","family":"Von Toussaint","sequence":"first","affiliation":[{"name":"Max-Planck-Institute for Plasmaphysics, Boltzmannstrasse 2, 85748 Garching, Germany"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2015,6,10]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"693","DOI":"10.1214\/07-BA228","article-title":"Nonparametric elicitation for heavy-tailed prior distributions","volume":"2","author":"Gosling","year":"2007","journal-title":"Bayesian Anal"},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"227","DOI":"10.1109\/TSSC.1968.300117","article-title":"Prior Probabilities","volume":"SSC4","author":"Jaynes","year":"1968","journal-title":"IEEE Trans. Syst. Sci. Cybern"},{"key":"ref_3","unstructured":"Kendall, M., and Moran, P. (1963). Geometrical Probability, Griffin."},{"key":"ref_4","doi-asserted-by":"crossref","unstructured":"Von der Linden, W., Dose, V., and von Toussaint, U. (2014). Bayesian Probability Theory: Application to the Physical Sciences, Cambridge University Press. [1st ed.].","DOI":"10.1017\/CBO9781139565608"},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"5356","DOI":"10.1364\/AO.43.005356","article-title":"Bayesian Neural-Networks-Based Evaluation of Binary Speckle Data","volume":"43","author":"Gori","year":"2004","journal-title":"Appl. Opt"},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"504","DOI":"10.1126\/science.1127647","article-title":"Reducing the Dimensionality of Data with Neural Networks","volume":"313","author":"Hinton","year":"2006","journal-title":"Science"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"529","DOI":"10.1038\/nature14236","article-title":"Human-level control through deep reinforcement learning","volume":"518","author":"Minh","year":"2015","journal-title":"Nature"},{"key":"ref_8","unstructured":"Williams, C.J. Hyperplane Priors. Bayesian Inference and Maximum Entropy Methods in Science and Engineering."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Box, G.E.P., and Tiao, G.C. (1992). Bayesian Inference in Statistical Analysis, Wiley. Reprint from 1973.","DOI":"10.1002\/9781118033197"},{"key":"ref_10","unstructured":"Zellner, A. (1971). An Introduction to Bayesian Inference in Econometrics, Wiley."},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"431","DOI":"10.1111\/j.2517-6161.1984.tb01317.x","article-title":"Outlier Models and Prior Distributions in Bayesian Linear Regression","volume":"46","author":"West","year":"1984","journal-title":"J. R. Stat. Soc. B"},{"key":"ref_12","unstructured":"O\u2019Hagan, A. (1994). Kendall\u2019s Advanced Theory of Statistics, Bayesian Inference, Arnold Publishers. [1st ed.]."},{"key":"ref_13","unstructured":"Landau, L., and Lifschitz, E. (1962). Lehrbuch der Theoretischen Physik I, Akademie Verlag. [1st ed.]."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/3898\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T20:47:43Z","timestamp":1760215663000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/17\/6\/3898"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2015,6,10]]},"references-count":13,"journal-issue":{"issue":"6","published-online":{"date-parts":[[2015,6]]}},"alternative-id":["e17063898"],"URL":"https:\/\/doi.org\/10.3390\/e17063898","relation":{},"ISSN":["1099-4300"],"issn-type":[{"type":"electronic","value":"1099-4300"}],"subject":[],"published":{"date-parts":[[2015,6,10]]}}}