Product partition models assume that observations in different components of a random partition of the data are independent. If the probability distribution of random partitions is in a certain product form prior to making the observations, it is also in product form given the observations. The product model thus provides a convenient machinery for allowing the data to weight the partitions likely to hold; and inference about particular future observations may then be made by first conditioning on the partition and then averaging over all partitions. These models apply with special computational simplicity to change point problems, where the partitions divide the sequence of observations into components within which different regimes hold. We show, with appropriate selection of prior product models, that the observations can eventually determine approximately the true partition.
"Product Partition Models for Change Point Problems." Ann. Statist. 20 (1) 260 - 279, March, 1992. https://doi.org/10.1214/aos/1176348521