A demanding challenge in Bayesian inversion is to efficiently characterize the posterior distribution. This task is problematic especially in high-dimensional non-Gaussian problems, where the structure of the posterior can be very chaotic and difficult to analyse. Often, in inverse problem literature one uses point estimators for this task. Here we discuss the maximum a posteriori (MAP) estimate, which is a computationally efficient method since it relates to an optimization problem. However, the scalability of the MAP estimate with respect to the discretization level has been an issue and in this talk we discuss its definition for infinite-dimensional problems. Moreover, we consider how Bregman distance can be used to characterize the MAP estimate. This is joint work with Martin Burger (University of Münster, Germany), Masoumeh Dashti (University of Sussex, UK) and Sergios Agapiou (University of Cyprus, Cyprus).

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