Q-Markov random probability measures and their posterior distributions

In this paper, we use the Markov property introduced in Balan and Ivanoff (J. Theoret. Probab. 15 (2002) 515) for set-indexed processes and we prove that a Markov prior distribution leads to a Markov posterior distribution. In particular, by proving that a neutral to the right prior distribution leads to a neutral to the right posterior distribution, we extend a fundamental result of Doksum (Ann. Probab. 2 (1974) 183) to arbitrary sample spaces.