It is not hard to show that if we use, say, the Bernoulli model class and the uniform prior and predict using the Bayes predictive distribution (i.e., Laplace's rule of succession), and the data is generated by a process that has indeed a stationary distribution, then with probability 1 our predictions will converge to the predictions that are optimal according to the stationary distribution although strings sampled from the stationary distribution are extremely hard to compute. In addition, the classical approach does not cost much. Hence sometimes (when the assumptions happen to be correct) it is preferrable, and it may also help us to do reasonable (though non-optimal) prediction, no matter whether the true distribution is computable or not. But unlike Laplace's approach the present one also yields asymptotically optimal predictions under the (strong but intriguing) assumption that the data is generated deterministically under certain resource constraints.
It will be of interest to
identify the precise set of priors dominated by 
.