General Hyperplane Prior Distributions Based on Geometric Invariances for Bayesian Multivariate Linear Regression

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.