In this paper, we focus on a type of inverse problem in which the data are 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 are 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 are computed. Inference is performed by taking posterior samples with adaptive MCMC. The method is illustrated on synthetic as well as real galactic data.