  Motivated by Bayesian inference with highly informative data we analyze the
performance of random walk-like Metropolis-Hastings algorithms for approximate
sampling of increasingly concentrating target distributions. We focus on
Gaussian proposals which use a Hessian-based approximation of the target
covariance. By means of pushforward transition kernels we show that for
Gaussian target measures the spectral gap of the corresponding
Metropolis-Hastings algorithm is independent of the concentration of the
posterior, i.e., the noise level in the observational data that is used for
Bayesian inference. Moreover, by exploiting the convergence of the
concentrating posteriors to their Laplace approximation we extend the analysis
to non-Gaussian target measures which either concentrate around a single point
or along a linear manifold. In particular, in that setting we show that the
average acceptance rate as well as the expected squared jump distance of
suitable Metropolis-Hastings Markov chains do not deteriorate as the target
concentrates.
