  We consider inverse problems in Hilbert spaces under correlated Gaussian
noise and use a Bayesian approach to find their regularised solution. We focus
on mildly ill-posed inverse problems with the noise being generalised
derivative of fractional Brownian motion, using a novel wavelet - based
approach we call vaguelette-vaguelette. It allows us to apply sequence space
methods without assuming that all operators are simultaneously diagonalisable.
The results are proved for more general bases and covariance operators.
  Our primary aim is to study the posterior contraction rate in such inverse
problems over Sobolev classes of true functions, comparing it to the derived
minimax rate. Secondly, we study the effect of plugging in a consistent
estimator of variances in sequence space on the posterior contraction rate, for
instance where there are repeated observations. This result is also applied to
the problem where the forward operator is observed with error. Thirdly, we show
that an adaptive empirical Bayes posterior distribution contracts at the
optimal rate, in the minimax sense, under a condition on prior smoothness, with
a plugged in maximum marginal likelihood estimator of the prior scale. These
theoretical results are illustrated on simulated data.
