A Fleming–Viot process and Bayesian nonparametrics

Walker, Stephen G.; Hatjispyros, Spyridon J.; Nicoleris, Theodoros

doi:10.1214/105051606000000600

Cited by 11 publications

(7 citation statements)

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“…Instead of inducing the temporal dependence through the building blocks of the stick-breaking representation (1.2), we let the dynamics of the dependent process be driven by a Fleming-Viot (FV) diffusion. FV processes have been extensively studied in relation to population genetics (see Ethier and Kurtz (1993) for a review), while their role in Bayesian nonparametrics was first pointed out in Walker et al (2007) (see also Favaro et al, 2009). A loose but intuitive way of thinking a FV diffusion is of being composed by infinitely-many probability masses, associated to different locations in the sampling space Y, each behaving like a diffusion in the interval [0, 1], under the overall constraint that the masses sum up to 1.…”

Section: Fleming-viot Dependent Dirichlet Processesmentioning

confidence: 99%

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Predictive inference with Fleming–Viot-driven dependent Dirichlet processes

2021

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We consider predictive inference using a class of temporally dependent Dirichlet processes driven by Fleming-Viot diffusions, which have a natural bearing in Bayesian nonparametrics and lend the resulting family of random probability measures to analytical posterior analysis. Formulating the implied statistical model as a hidden Markov model, we fully describe the predictive distribution induced by these Fleming-Viot-driven dependent Dirichlet processes, for a sequence of observations collected at a certain time given another set of draws collected at several previous times. This is identified as a mixture of Pólya urns, whereby the observations can be values from the baseline distribution or copies of previous draws collected at the same time as in the usual Pòlya urn, or can be sampled from a random subset of the data collected at previous times. We characterise the time-dependent weights of the mixture which select such subsets and discuss the asymptotic regimes. We describe the induced partition by means of a Chinese restaurant process metaphor with a conveyor belt, whereby new customers who do not sit at an occupied table open a new table by picking a dish either from the baseline distribution or from a time-varying offer available on the conveyor belt. We lay out explicit algorithms for exact and approximate posterior sampling of both observations and partitions, and illustrate our results on predictive problems with synthetic and real data.

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Section: Fleming-viot Dependent Dirichlet Processesmentioning

confidence: 99%

“…The transition function that characterizes a FV process admits the following natural interpretation in Bayesian nonparametrics (cf. Walker et al, 2007). Initiate the process at the RPM X 0 ∼ Π α , and denote by D t a time-indexed latent variable taking values in Z + .…”

Section: Fleming-viot Dependent Dirichlet Processesmentioning

confidence: 99%

Predictive inference with Fleming–Viot-driven dependent Dirichlet processes

2021

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“…This more principled approach to building time series with given marginals has been well explored, both probabilistically and statistically, for finite-dimensional marginal distributions, either using processes with discontinuous sample paths, as in Barndorff-Nielsen and Shephard (2001) or Griffin (2011), or using diffusions, as we undertake here. The relevance of measurevalued diffusions in Bayesian nonparametrics has been pioneered in Walker et al (2007), whose construction naturally allows for separate control of the marginal distributions and the memory of the process.…”

Section: Motivation and Main Contributionsmentioning

confidence: 99%

“…It follows that, using terms familiar to the Bayesian literature, under this parametrisation the FV can be considered as a dependent Dirichlet process with continuous sample paths. Constructions of Fleming-Viot and closely related processes using ideas from Bayesian nonparametrics have been proposed in Walker et al (2007); Favaro, Ruggiero and Walker (2009); Ruggiero and Walker (2009a;b). Different classes of diffusive dependent Dirichlet processes or related are constructed in Mena and Ruggiero (2016); Mena et al (2011) based on the stick-breaking representation (Sethuraman, 1994).…”

Section: The Fleming-viot Processmentioning

confidence: 99%

Conjugacy properties of time-evolving Dirichlet and gamma random measures

Papaspiliopoulos¹,

Ruggiero²,

Spanò³

2016

Electron. J. Statist.

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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.

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“…Measure-valued Markov chains, or more generally measure-valued Markov processes, arise naturally in modeling the composition of evolving populations and play an important role in a variety of research areas such as population genetics and bioinformatics (see, e.g., [5,9,10,26]), Bayesian nonparametrics [31,38], combinatorics [26] and statistical physics [5,6,26]. In particular, in Bayesian nonparametrics there has been interest in measure-valued Markov chains since the seminal paper by [12], where the law of the Dirichlet process has been characterized as the unique invariant measure of a certain measure-valued Markov chain.…”

Section: Introductionmentioning

confidence: 99%

A class of measure-valued Markov chains and Bayesian nonparametrics

Favaro¹,

Guglielmi²,

Walker³

2012

Bernoulli

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Measure-valued Markov chains have raised interest in Bayesian nonparametrics since the seminal paper by (Math. Proc. Cambridge Philos. Soc. 105 (1989) 579--585) where a Markov chain having the law of the Dirichlet process as unique invariant measure has been introduced. In the present paper, we propose and investigate a new class of measure-valued Markov chains defined via exchangeable sequences of random variables. Asymptotic properties for this new class are derived and applications related to Bayesian nonparametric mixture modeling, and to a generalization of the Markov chain proposed by (Math. Proc. Cambridge Philos. Soc. 105 (1989) 579--585), are discussed. These results and their applications highlight once again the interplay between Bayesian nonparametrics and the theory of measure-valued Markov chains.Comment: Published in at http://dx.doi.org/10.3150/11-BEJ356 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

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A Fleming–Viot process and Bayesian nonparametrics

Cited by 11 publications

References 11 publications

Predictive inference with Fleming–Viot-driven dependent Dirichlet processes

Predictive inference with Fleming–Viot-driven dependent Dirichlet processes

Conjugacy properties of time-evolving Dirichlet and gamma random measures

A class of measure-valued Markov chains and Bayesian nonparametrics

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