Intrinsic Bayesian Errors-in-Variables Under Anisotropic Noise

We present an intrinsic Bayesian formulation of the linear errors-in-variables problem under anisotropic Gaussian noise using a homogeneous representation. By analytically marginalizing latent noise-free data, a closed-form marginal likelihood is obtained in arbitrary dimension. Under an ellipsoidal normalization, the resulting maximum likelihood estimator coincides with generalized total least squares. Bayesian inference is developed on an affine chart via homogenization, yielding an explicit posterior decomposition into a heavy-tailed radial component and an affine Gaussian term with a chart-dependent correction factor, and ensuring posterior properness under a Gaussian prior. A conditionally conjugate variational approximation enables efficient inference and admits a recursive implementation with exponential forgetting, resulting in a practical online variational EIV algorithm with closed-form updates. The method is illustrated on an application-motivated melt-flow-rate identification problem from plastic recycling, with anisotropic measurement noise and direct comparison to a numerically computed reference posterior. The example is used to assess recursive tracking and posterior-approximation quality under slight model misspecification.