Measure-valued stochastic processes

Moving masses

October 16, 2020 — May 3, 2022

functional analysis
kernel tricks
machine learning
point processes
stochastic processes
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 exhaustingly 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 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

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.