A stochastic ordering and related sequential decision problems

For sequential stochastic decision problems on a partially observable Markov process, several properties under Bayesian learning procedure are considered. Under several assumptions and a partial order defined for the probability measures on the state space, it is shown that the Bayesian posterior distribution of the state of the Markov process is monotonic with respect to the observation of the random variable and the prior distribution in the sense of an order defined here. Finally, these results and additional properties are adapted to some sequential decision problems on this partially observable Markov process.