  We propose a dimension reduction technique for Bayesian inverse problems with
nonlinear forward operators, non-Gaussian priors, and non-Gaussian observation
noise. The likelihood function is approximated by a ridge function, i.e., a map
which depends non-trivially only on a few linear combinations of the
parameters. We build this ridge approximation by minimizing an upper bound on
the Kullback--Leibler divergence between the posterior distribution and its
approximation. This bound, obtained via logarithmic Sobolev inequalities,
allows one to certify the error of the posterior approximation. Computing the
bound requires computing the second moment matrix of the gradient of the
log-likelihood function. In practice, a sample-based approximation of the upper
bound is then required. We provide an analysis that enables control of the
posterior approximation error due to this sampling. Numerical and theoretical
comparisons with existing methods illustrate the benefits of the proposed
methodology.
