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We consider a Bayesian nonparametric approach to a family of linear inverseproblems in a separable Hilbert space setting with Gaussian noise. We assumeGaussian priors, which are conjugate to the model, and present a method ofidentifying the posterior using its precision operator. Working with theunbounded precision operator enables us to use partial differential equations(PDE) methodology to obtain rates of contraction of the posterior distributionto a Dirac measure centered on the true solution. Our methods assume arelatively weak relation between the prior covariance, noise covariance andforward operator, allowing for a wide range of applications.

Posterior Contraction Rates for the Bayesian Approach to Linear Ill-Posed Inverse Problems