Conjugacy properties of time-evolving Dirichlet and gamma random measures

Abstract

We extend classic characterisations of posterior distributions under Dirichlet process and gamma random measures priors to a dynamic framework. We consider the problem of learning, from indirect observations, two families of time-dependent processes of interest in Bayesian nonparametrics: the first is a dependent Dirichlet process driven by a Fleming-Viot model, and the data are random samples from the process state at discrete times; the second is a collection of dependent gamma random measures driven by a Dawson-Watanabe model, and the data are collected according to a Poisson point process with intensity given by the process state at discrete times. Both driving processes are diffusions taking values in the space of discrete measures whose support varies with time, and are stationary and reversible with respect to Dirichlet and gamma priors respectively. A common methodology is developed to obtain in closed form the time-marginal posteriors given past and present data. These are shown to belong to classes of finite mixtures of Dirichlet processes and gamma random measures for the two models respectively, yielding conjugacy of these classes to the type of data we consider. We provide explicit results on the parameters of the mixture components and on the mixing weights, which are time-varying and drive the mixtures towards the respective priors in absence of further data. Explicit algorithms are provided to recursively compute the parameters of the mixtures. Our results are based on the projective properties of the signals and on certain duality properties of their projections.

Predictive inference with Fleming–Viot-driven dependent Dirichlet processes

“…The continuous time dependence is a distinctive feature of our proposal, compared to the bulk of literature in the area. In particular, here we complement previous work done in Papaspiliopoulos et al (2016), which focussed on identifying the laws of the dependent RPMs involved, by investigating the distributional properties of future observations, characterized as a mixture of Pólya urn schemes, and those of the induced partitions.…”

“…The signal can be thought of as the discrete-time sampling of a continuous time process, and is assumed to completely specify the distributions of the observations, called emission distributions. While the literature on hidden Markov models has mainly focussed on finite-dimensional signals, infinite-dimensional cases have been previously considered in Beal et al (2002), Van Gael et al (2008), Stepleton et al (2009), Yau et al (2011), Zhang et al (2014, Papaspiliopoulos et al (2016).…”

Exact inference for a class of hidden Markov models on general state spaces

“…These conditions are formulated for generic K-dimensional HMMs, and the implied computations do not rely on specific properties of the model at hand (Chaleyat-Maurel and Genon-Catalot (2009) for example exploit the specific eigenstructure of the transition semigroup). Later, Papaspiliopoulos et al (2016) showed computable filtering is possible for two signals taking values on the space of atomic measures, namely Fleming-Viot and Dawson-Watanabe measure-valued diffusions, and more recently Ascolani et al (2020) applied their results to Bayesian predictive inference in a nonparametric framework.…”

“…The present article obtains theory and practical algorithms for the whole inferential agenda for HMMs. While Papaspiliopoulos and Ruggiero (2014); Papaspiliopoulos et al (2016) focused only on the theoretical aspects of filtering, here we develop and implement results also on smoothing and inference on the model parameters. Under a set of sufficient conditions essentially analogous to those in Papaspiliopoulos and Ruggiero (2014), we show that all distributions of interest for the signal can be expressed as finite mixtures of elementary densities, and the likelihood of the observations takes the form of a finite product of mixtures.…”

Clustering dynamics in a class of normalised generalised gamma dependent priors

“…Scaling limits of the clustering dynamics for other classes of dependent models connected with Bayesian nonparametrics have been studied in Ruggiero et al (2013); Ruggiero (2014) for the normalised inverse gaussian and the two-parameter Poisson-Dirichlet case, respectively. See also Ruggiero and Walker (2009a;; Mena et al (2011); Mena and Ruggiero (2016); Papaspiliopoulos et al (2016) for different dependent models connected with diffusions processes.…”