Proof of The TAP Free Energy for High-Dimensional Linear Regression with Spherical Priors at All Temperatures
全温度下球面先验高维线性回归的 TAP 自由能证明
Approximate inference is central to Bayesian learning, with variational inference (VI) providing a scalable framework for posterior approximation. While mean-field VI often fails in high dimensions, the more refined Bethe approximation, equivalent to the Thouless-Anderson-Palmer (TAP) free energy in statistical physics, has long been conjectured to capture Bayes-optimal behavior. We prove that the TAP formula holds for Bayesian linear regression with a uniform spherical…