# Measure-valued stochastic processes

Moving masses

October 16, 2020 — May 3, 2022

functional analysis

kernel tricks

machine learning

nonparametric

PDEs

physics

point processes

probability

regression

SDEs

spatial

statistics

stochastic processes

uncertainty

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

## 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*.