Let us consider the composite system of the N+M identical quantum states, each of which is specified by the unknown finite-dimensional parameter. Compared to usual quantum estimation or tomography scheme, we assume that we have only N+M resources available. Our task is to perform the measurement on the M subsystems and to predict the expectation of an arbitrary chosen observable (spin, momentum etc.) in the remaining N systems "after" obtaining the measurement outcome.
In the above situation, we show that the Bayesian predictive method in statistics is very useful. Some optimality results and simple examples are presented.