  It is well known that a Bayesian probability forecast for all future
observations should be a probability measure in order to satisfy a natural
condition of coherence. The main topics of this paper are the evolution of the
Bayesian probability measure and ways of testing its adequacy as it evolves
over time. The process of testing evolving Bayesian beliefs is modelled in
terms of betting, similarly to the standard Dutch book treatment of coherence.
The resulting picture is adapted to forecasting several steps ahead and making
almost optimal decisions.
