Bayesian Inverse Optimisation under Uncertainty Posterior Inference for Hidden Decision Objectives

Inverse optimization seeks the hidden objective that explains observed decisions, yet existing methods typically recover a single objective and ignore uncertainty in the underlying preferences. We develop a Bayesian framework that represents the unknown objective by a posterior distribution, combining a boundedrationality choice model with prior information to quantify uncertainty directly. We establish geometric conditions for identifiability, prove posterior consistency, derive the optimal n−1/2, contraction rate, obtain a Bernstein–von Mises approximation, and present a finite-sample concentration bound together with a robust fractional posterior for misspecified behavioural models. Posterior inference is performed using gradient-based Markov chain Monte Carlo and a Laplace approximation. Simulation studies demonstrate accurate recovery under noisy, misspecified, nonlinear, and high-dimensional settings, while a controlled clinical example illustrates uncertainty quantification in hidden decision preferences. The proposed framework extends inverse optimization from point estimation to statistically principled uncertainty quantification, providing a rigorous foundation for learning decision objectives from imperfect observations.