Bayesian inverse problems confront uncertain parameters by combining observations with a forward model to produce a posterior distribution. When the forward model itself is imperfect or nonlinear model error is present, naive application of Bayes leads to biased estimates and overconfident uncertainty. Andrew M. Stuart at California Institute of Technology emphasizes rigorous probabilistic formulation for inverse problems to ensure well-posedness and meaningful uncertainty quantification. Addressing nonlinear model error requires treating the discrepancy between the true physical process and the computational model as an additional uncertain quantity rather than a fixed bias.
Modeling nonlinear error within Bayesian inversion
A practical route embeds model discrepancy as a stochastic term inside the forward model and treats it as a latent field to be inferred jointly with parameters. Tony O'Hagan at University of Sheffield advocated using flexible priors such as Gaussian process priors to represent systematic discrepancy in computer model calibration. For nonlinear error, this prior must allow non-Gaussian features or be combined with basis expansions that capture state-dependent biases. Hierarchical Bayesian models then place priors on hyperparameters that control amplitude and smoothness, enabling the posterior to learn where the model is unreliable and to propagate that uncertainty into parameter estimates.
Computational strategies and consequences
Computationally, incorporating nonlinear model error increases dimension and nonlinearity, so methods like Markov chain Monte Carlo with surrogate models, variational approximations, and transport map approaches developed by Youssef Marzouk at Massachusetts Institute of Technology can be effective. Emulation reduces cost while explicitly modeling discrepancy, but careful validation is essential because poor emulators can hide structural errors. Failure to account for nonlinear model error can mislead decisions in applied domains. In geophysical inversion for groundwater or seismic hazard, underestimated uncertainty affects policy, infrastructure planning, and community safety. In environmental modeling, regional biases tied to terrain or land use produce territorial consequences when models trained on one area are applied elsewhere.
Treating model error probabilistically also has cultural and institutional implications. Transparent discrepancy modeling builds trust with stakeholders by demonstrating where confidence is low, which is critical for public acceptance of policy based on uncertain models. Finally, theoretical work shows that correctly specified hierarchical models can recover consistent posteriors as data accumulate, whereas misspecification generally yields persistent bias. Practical practice therefore combines principled probabilistic modeling, diagnostic checks against withheld data, and engagement with domain experts to design discrepancy priors that reflect physical understanding and social context.