On a prediction problem for an infinite dimensional hidden Markov model

We consider the problem of filtering the latent state of a hidden Markov model which is assumed infinite-dimensional and to be a Fleming-Viot diffusion, which induces a prior distribution on the space of continuous functions from the positive half line to the space of discrete probability measures. While updating the conditional law of the latent state given the incoming observations is straightforward, in virtue of the conjugacy property of the Dirichlet process, prediction involves infinite sums, even for the computation of certain “moments”. Here we show how the problem can be overcome analytically, avoiding the need for approximation or simulation techniques.