Low Rank Independence Samplers in Bayesian Inverse Problems
For practitioners of Bayesian inverse problems, this method reduces computational cost of MCMC sampling while maintaining accuracy.
The paper presents a computationally efficient Metropolis-Hastings independence sampler for high-dimensional Gaussian posteriors in Bayesian linear inverse problems, using a low-rank proposal distribution. Numerical experiments show high acceptance rates and applicability to image deblurring, tomography, and NMR relaxometry.
In Bayesian inverse problems, the posterior distribution is used to quantify uncertainty about the reconstructed solution. In practice, Markov chain Monte Carlo algorithms often are used to draw samples from the posterior distribution. However, implementations of such algorithms can be computationally expensive. We present a computationally efficient scheme for sampling high-dimensional Gaussian distributions in ill-posed Bayesian linear inverse problems. Our approach uses Metropolis-Hastings independence sampling with a proposal distribution based on a low-rank approximation of the prior-preconditioned Hessian. We show the dependence of the acceptance rate on the number of eigenvalues retained and discuss conditions under which the acceptance rate is high. We demonstrate our proposed sampler by using it with Metropolis-Hastings-within-Gibbs sampling in numerical experiments in image deblurring, computerized tomography, and NMR relaxometry.