Nonparametric Bayesian inverse problems with Laplacian noise
The focus of this work are Bayesian inverse problems in an infinite-dimensional setting with Gaussian prior and data corrupted by additive Laplacian noise. In particular, the connection between Tikhonov–Phillips regularisation with an l^1-discrepancy term and Bayesian MAP estimation based upon a Laplacian noise model is investigated for linear problems as well as the consistency of MAP and CM estimator. For Laplacian infinite product measures on a separable Hilbert space, a result similar to the Cameron–Martin theorem for Gaussian measures is shown, stating the space of admissible shifts and the relative density of shifted Laplacian measures. Under certain conditions on the log-likelihood, MAP estimates in separable Hilbert spaces are characterised as minimisers of the Onsager–Machlup functional of the posterior distribution, which in this case has the form of a Tikhonov–Phillips functional with a discrepancy term given by the log-likehood and a squared norm penalty term. The behaviour of MAP and CM estimator is studied for a severely ill-posed linear problem; a generalised form of the inverse heat equation, under the presence of additive Laplacian noise. The posterior distribution is derived via Bayesian inference and both MAP and CM estimator are computed explicitly. The MAP estimator is shown to be asymptotically unbiased in a frequentist setting. An estimate for the convergence rate of the bias is stated under an analytic source condition. Moreover, an estimate for the convergence rate of the mean squared error of the MAP estimator is proved under an analytic source condition and in conjunction with an a priori parameter choice. This rate is then compared to the minimax rate in the fully Gaussian case. The behaviour and consistency of MAP and CM estimator is studied numerically for the classical inverse heat equation in one dimension with additive Laplacian noise. The empirical MSE of both estimators is observed to converge to zero in the small noise limit with the estimated rate if an analytic source condition is satisfied, whereas neither MAP nor CM estimator converge towards the true solution in mean square if only a Sobolev-type source condition is satisfied. Moreover, empirical confidence regions around both estimators are computed. Finally, a direct sampler for the posterior distribution is developed and used to compute credible regions around both estimators.