  We consider a class of linear ill-posed inverse problems arising from
inversion of a compact operator with singular values which decay exponentially
to zero. We adopt a Bayesian approach, assuming a Gaussian prior on the unknown
function. If the observational noise is assumed to be Gaussian then this prior
is conjugate to the likelihood so that the posterior distribution is also
Gaussian. We study Bayesian posterior consistency in the small observational
noise limit. We assume that the forward operator and the prior and noise
covariance operators commute with one another. We show how, for given
smoothness assumptions on the truth, the scale parameter of the prior can be
adjusted to optimize the rate of posterior contraction to the truth, and we
explicitly compute the logarithmic rate.
