  In this paper we focus on a type of inverse problem in which the data is
expressed as an unknown function of the sought and unknown model function (or
its discretised representation as a model parameter vector). In particular, we
deal with situations in which training data is not available. Then we cannot
model the unknown functional relationship between data and the unknown model
function (or parameter vector) with a Gaussian Process of appropriate
dimensionality. A Bayesian method based on state space modelling is advanced
instead. Within this framework, the likelihood is expressed in terms of the
probability density function ($pdf$) of the state space variable and the sought
model parameter vector is embedded within the domain of this $pdf$. As the
measurable vector lives only inside an identified sub-volume of the system
state space, the $pdf$ of the state space variable is projected onto the space
of the measurables, and it is in terms of the projected state space density
that the likelihood is written; the final form of the likelihood is achieved
after convolution with the distribution of measurement errors. Application
motivated vague priors are invoked and the posterior probability density of the
model parameter vectors, given the data is computed. Inference is performed by
taking posterior samples with adaptive MCMC. The method is illustrated on
synthetic as well as real galactic data.
