  For ill-posed inverse problems, a regularised solution can be interpreted as
a mode of the posterior distribution in a Bayesian framework. This framework
enriches the set the solutions, as other posterior estimates can be used as a
solution to the inverse problem, such as the posterior mean that can be easier
to compute in practice. In this paper we prove consistency of Bayesian
solutions of an ill-posed linear inverse problem in the Ky Fan metric for a
general class of likelihoods and prior distributions in a finite dimensional
setting. This result can be applied to study infinite dimensional problems by
letting the dimension of the unknown parameter grow to infinity which can be
viewed as discretisation on a grid or spectral approximation of an infinite
dimensional problem. Likelihood and the prior distribution are assumed to be in
an exponential form that includes distributions from the exponential family,
and to be differentiable. The observations can be dependent. No assumption of
finite moments of observations, such as expected value or the variance, is
necessary thus allowing for possibly non-regular likelihoods, and allowing for
non-conjugate and improper priors. If the variance exists, it may be
heteroscedastic, namely, it may depend on the unknown function. We observe
quite a surprising phenomenon when applying our result to the spectral
approximation framework where it is possible to achieve the parametric rate of
convergence, i.e the problem becomes self-regularised. We also consider a
particular case of the unknown parameter being on the boundary of the parameter
set, and show that the rate of convergence in this case is faster than for an
interior point parameter.
