The objective of stochastic modelling is not to find an exact representation of the observed data itself, but rather to build a statistical model of the process which generates the data. In contrast to the frequentist approach, the Bayesian approach provides a different framework for dealing with the issues of model complexity and, thus, avoiding the overfitting problem. The objective of this paper is to adopt the Bayesian framework for modelling the long memory of foreign currency exchange rate volatilities on an operational time scale in the context of the ARCH methodology. In addition, it is illustrated that the choice of a prior distribution for the model parameters affects the overall model quality. Even though attention is focussed on ARCH models, the suggested approach can be applied to other model types too.