  Markov chain Monte Carlo (MCMC) methods form one of the algorithmic
foundations of Bayesian inverse problems. The recent development of
likelihood-informed subspace (LIS) methods offers a viable route to designing
efficient MCMC methods for exploring high-dimensional posterior distributions
via exploiting the intrinsic low-dimensional structure of the underlying
inverse problem. However, existing LIS methods and the associated performance
analysis often assume that the prior distribution is Gaussian. This assumption
is limited for inverse problems aiming to promote sparsity in the parameter
estimation, as heavy-tailed priors, e.g., Laplace distribution or the elastic
net commonly used in Bayesian LASSO, are often needed in this case. To overcome
this limitation, we consider a prior normalization technique that transforms
any non-Gaussian (e.g. heavy-tailed) priors into standard Gaussian
distributions, which makes it possible to implement LIS methods to accelerate
MCMC sampling via such transformations. We also rigorously investigate the
integration of such transformations with several MCMC methods for
high-dimensional problems. Finally, we demonstrate various aspects of our
theoretical claims on two nonlinear inverse problems.
