Smoothing inference for Fleming-Viot-driven dependent Dirichlet Processes

Fleming-Viot-driven dependent Dirichlet Processes (FVDDP) are a class of Hidden Markov Models that consist of a time-dependent finite mixture of Dirichlet Processes controlled by a Fleming-Viot process. Although this model has its origin in the study of population genetics, it is possible to exploit its predictive structure to formulate various types of inference. In fact it is possible to determine the unobserved states of this process and to compute their evolution over time, which is related to the law of a multidimensional death process, and their update with respect to the collected data, linked to the Pólya Urn sampling scheme. This process lends itself to be characterized in terms of groups of active observations at a certain time. In this regard, the study of smoothing distribution is worthy of great interest, which allows the estimation of the state of the signal given observations collected at earlier and later times with respect to the one of interest, giving special emphasis to types observed at multiple collection times. This work introduces notions related to Hidden Markov Models, Bayesian Nonparametrics, and Duality for Markov processes, which underlie the study of FVDDP, and aims to summarize the main theoretical results on this topic. In addition, emphasis is given to the computational approach, introducing algorithms, commenting on their properties, the details of implementation, and the performance on synthetic data. This has led to the creation of an R package, FVDDPpkg, which can be found on GitHub, allowing the user to easily dispose of methods related to Fleming-Viot dependent Dirichlet Processes.