Measure-valued stochastic processes

Moving masses

2020-10-16 — 2022-05-02

Wherein Measure-Valued Stochastic Processes Are Examined, and a Dependent Dirichlet Process Is Described in Which Stick-Breaking Weights Are Given by Transforms of a Stochastic Process to Induce Indexwise Dependence.

functional analysis
kernel tricks
machine learning
nonparametric
PDEs
physics
point processes
probability
regression
SDEs
spatial
statistics
stochastic processes
uncertainty
Figure 1

Measure-valued random variates with some kind of dependence with respect to an index.

1 Dependent Dirichlet process

Invented by Steven N. MacEachern (1999), this idea produced a whole family of related models, reviewed exhaustively in Quintana et al. (2022) and Foti and Williamson (2015). The idea is to construct correlation amongst the defining RVs of a Dirichlet process in the stick-breaking construction by declaring them to be given as some transform of a stochastic process. Posterior inference for these models does not appear to be especially nice; I would like a nice reference for that.

2 Discrete case

See discrete measure processes.

3 References

Blackwell, and MacQueen. 1973. “Ferguson Distributions Via Polya Urn Schemes.” The Annals of Statistics.
Foti, and Williamson. 2015. “A Survey of Non-Exchangeable Priors for Bayesian Nonparametric Models.” IEEE Transactions on Pattern Analysis and Machine Intelligence.
Gelfand, Kottas, and MacEachern. 2005. “Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing.” Journal of the American Statistical Association.
Ishwaran, and James. 2001. “Gibbs Sampling Methods for Stick-Breaking Priors.” Journal of the American Statistical Association.
MacEachern, Steven N. 1999. “Dependent Nonparametric Processes.” In ASA Proceedings of the Section on Bayesian Statistical Science.
MacEachern, Steven N, Kottas, and Gelfand. 2001. “Spatial Nonparametric Bayesian Models.”
Moraffah, and Papandreou-Suppappola. 2022. “Bayesian Nonparametric Modeling for Predicting Dynamic Dependencies in Multiple Object Tracking.” Sensors.
Nieto-Barajas, Prünster, and Walker. 2004. “Normalized Random Measures Driven by Increasing Additive Processes.” Annals of Statistics.
Quintana, Müller, Jara, et al. 2022. “The Dependent Dirichlet Process and Related Models.” Statistical Science.