Constraint residuals, graph posteriors, and determinant-corrected full-space targets in Bayesian inverse problems

Bayesian inverse problems constrained by state equations are often sampled in a full parameter-state space by penalising the residual, rather than in a reduced space where the state is eliminated. We show that these formulations are not automatically equivalent as posterior measures. For finite-dimensional discretisations of equality-constrained inverse problems, assume the state equation \(c(\theta,u)=0\) has a unique solution \(u=G(\theta)\) and nonsingular state Jacobian \(\D_u c\). The reduced posterior, its graph lift, and the zero-noise residual posterior are then distinct. A local change of variables shows that an uncorrected Gaussian residual penalty converges, after marginalisation over \(u\), to the reduced density multiplied by \(\abs{\det \D_u c(\theta,G(\theta))}^{-1}\). Thus algebraically equivalent residuals can define the same feasible set but different limiting posteriors. We derive determinant corrections for unweighted, weighted, and rescaled residual penalties that have the graph-lifted reduced posterior as their hard-constraint limit. The result separates feasibility from posterior calibration: driving the residual to zero is not sufficient for exact sampling of the graph-lifted reduced posterior unless the sampling or correction step targets the corresponding corrected density.