Bayesian inverse problems with Laplacian noise We are interested in Bayesian inverse problems on function spaces with non-Gaussian noise. In the case of Gaussian noise and a Gaussian prior, MAP estimators can be characterised as minimisers of the Onsager-Machlup functional. We show that this connection also holds true for Laplacian noise. It provides a rigorous derivation of variational regularisations based upon explicit assumptions. Subsequently, we use this knowledge to study the inverse heat equation in a Bayesian setting with Laplacian noise and Gaussian prior. We make sure that a solution to the Bayesian inverse problem exists, determine its Onsager-Machlup functional and prove that the MAP estimator is consistent in a frequentist sense.