  In this work the issue of Bayesian inference for stationary data is
addressed. Therefor a parametrization of a statistically suitable subspace of
the the shift-ergodic probability measures on a Cartesian product of some
finite state space is given using an inverse limit construction. Moreover, an
explicit model for the prior is given by taking into account an additional step
in the usual stepwise sampling scheme of data. An update to the posterior is
defined by exploiting this augmented sample scheme. Thereby, its model-step is
updated using a measurement of the empirical distances between the model
classes.
