Past exam of the advanced topics in statistical machine learning course of the University of Oxford / 2025 / 1 / a / iii / Solution

Evaluate Bayes' theorem at the posterior mode $\mu$:The multivariate normal density in the Laplace approximation has height $(2\pi {)}^{-d/2}|\Sigma {|}^{-1/2}$ at its mean. ThereforeEquivalently, the approximate log marginal likelihood isThe determinant factor accounts for the volume of the locally plausible parameter region, which the peak likelihood alone does not measure.