Mixed Least Squares--Total Least Squares from a Bayesian Perspective

Real-time estimation of partial error-in-variables (EIV) models is formulated within a Bayesian framework. The underlying regression-type stochastic model accounts for inaccurate measurements in selected regressor components corrupted by additive normal noise. Since existing partial EIV solvers are often computationally demanding and provide only point estimates, we derive a marginal output density and project the resulting posterior kernel onto the normal-Wishart distribution using a fixed-form variational approximation. The optimized posterior mean recovers the mixed least-squares-total-least-squares (LS-TLS) estimate and relates it to the maximum-likelihood (ML) solution. The paper also derives the exact Student's t marginal posterior of the regression coefficients, establishes Fisher consistency of the Rayleigh-like mixed LS-TLS criterion, and contrasts its population target with that of ordinary LS, which is biased in the partial EIV setting.