Bayesian Inference in Inverse Problems

Inverse problems arise when one seeks to recover unknown parameters or functions from indirect, noisy observations via a forward model. The Bayesian framework casts this recovery as the updating of a prior probability distribution over the unknowns into a posterior distribution that quantifies remaining uncertainty. The prior encodes information or belief about smoothness, sparsity or physical constraints, while the likelihood measures consistency between model predictions and data. The posterior thus provides a principled regularisation, avoiding overfitting and offering error bars for predictive quantities of interest. In many applications – from medical imaging and geophysical tomography to climate modelling and material characterisation – parameters live in infinite-dimensional spaces and must be discretised for computation. This raises challenges of mesh-independence, high dimensionality and multimodality. Modern algorithms address these challenges through advanced sampling schemes, Gaussian approximations, hierarchical and non-Gaussian priors, and scalable inference tools. Overall, Bayesian inverse problems unify data assimilation and uncertainty quantification within a coherent probabilistic paradigm, enabling robust parameter estimation and decision making under uncertainty.

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Recent advances in sampling infinite-dimensional posteriors have combined geometric insights with robust discretisation techniques. A novel class of Markov chain Monte Carlo methods exploits intrinsic low-dimensional subspaces informed by data, yielding mesh-independent convergence rates while preserving efficiency in the complement space. These methods have demonstrated order-of-magnitude gains in sampling efficiency for subsurface flow and heat conduction problems.

Alternative efforts have focused on Laplace-based numerical integration for highly concentrated posteriors. By centring Gaussian proposals at the maximum a posteriori estimate and employing local curvature information, importance sampling and quasi-Monte Carlo methods become robust to diminishing observational noise. Such approaches maintain efficiency where prior-based schemes fail, facilitating accurate estimation of posterior means and variances in moderate to high dimensions.

On the modelling front, non-stationary Matérn priors with stochastic partial differential equation representations have been endowed with hyperpriors to tune local correlation lengths. This hierarchical scheme balances smoothness and edge preservation in deconvolution and interpolation tasks, adapting automatically to spatial heterogeneity. The result is a flexible inversion framework that captures sharp interfaces without oversmoothing, while delivering posterior uncertainty quantification via efficient Gibbs and Metropolis-within-Gibbs sampling.

Bayesian Inference in Inverse Problems publication trend

The graph below shows the total number of articles in bayesian inference in inverse problems across all publications each year (not limited to Nature Index journals).

Technical terms

Inverse problem: The task of determining unknown model inputs from observed outputs via an often ill-posed forward map.

Prior distribution: A probability measure expressing beliefs about unknowns before observing data.

Likelihood function: A function quantifying the plausibility of observed data given model parameters.

Posterior distribution: The updated probability measure over unknowns after combining prior and likelihood via Bayes’s theorem.

Regularisation: The process of stabilising an ill-posed problem by incorporating additional information, often implicit in the prior.

Markov chain Monte Carlo (MCMC): A family of algorithms for drawing dependent samples from complex posterior distributions.

Laplace approximation: A Gaussian approximation of the posterior obtained by expanding the log-posterior about its mode.

Hyperprior: A prior placed on parameters of another prior distribution, enabling hierarchical Bayesian modelling.

References

1. Geometric MCMC for infinite-dimensional inverse problems. Journal of Computational Physics (2017).
2. On the convergence of the Laplace approximation and noise-level-robustness of Laplace-based Monte Carlo methods for Bayesian inverse problems. Numerische Mathematik (2020).
3. Hyperpriors for Matérn fields with applications in Bayesian inversion. Inverse Problems and Imaging (2019).