  We consider the problem of learning two families of time-evolving random
measures from indirect observations. In the first model, the signal is a
Fleming--Viot diffusion, which is reversible with respect to the law of a
Dirichlet process, and the data is a sequence of random samples from the state
at discrete times. In the second model, the signal is a Dawson--Watanabe
diffusion, which is reversible with respect to the law of a gamma random
measure, and the data is a sequence of Poisson point configurations whose
intensity is given by the state at discrete times. A common methodology is
developed to obtain the filtering distributions in a computable form, which is
based on the projective properties of the signals and duality properties of
their projections. The filtering distributions take the form of mixtures of
Dirichlet processes and gamma random measures for each of the two families
respectively, and an explicit algorithm is provided to compute the parameters
of the mixtures. Hence, our results extend classic characterisations of the
posterior distribution under Dirichlet process and gamma random measures priors
to a dynamic framework.
