  In this paper we study properties of the Laplace approximation of the
posterior distribution arising in nonlinear Bayesian inverse problems. Our work
is motivated by Schillings et al. (2020), where it is shown that in such a
setting the Laplace approximation error in Hellinger distance converges to zero
in the order of the noise level. Here, we prove novel error estimates for a
given noise level that also quantify the effect due to the nonlinearity of the
forward mapping and the dimension of the problem. In particular, we are
interested in settings in which a linear forward mapping is perturbed by a
small nonlinear mapping. Our results indicate that in this case, the Laplace
approximation error is of the size of the perturbation. The paper provides
insight into Bayesian inference in nonlinear inverse problems, where
linearization of the forward mapping has suitable approximation properties.
