The current thesis consists of two results obtained during my PhD, both related to approximations of high/infinite-dimensional measures emerging from the Bayesian approach to inverse problems. In the first part, we study a technique for the reduction of the dimension in the finite but high-dimensional case when the prior is 1-exponentially distributed. In Chapter 4, this is done in a way that the approximated posterior measure minimises the distance to the posterior by using an appropriate metric. In the second part, we consider the problem of estimating the drift and diffusion coefficient of a stochastic differential equation using noisy measurements on a single path. There, we use a perturbation technique on the solution of the SDE to obtain an approximated posterior; in Chapter 5, we study the convergence properties of this approximation
