Measure-valued random variates with some kind of dependence with respect to an index.
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.
Discrete case
References
Foti, Nicholas J., and Sinead A. Williamson. 2015. “A Survey of Non-Exchangeable Priors for Bayesian Nonparametric Models.” IEEE Transactions on Pattern Analysis and Machine Intelligence 37 (2): 359–71.
Gelfand, Alan E, Athanasios Kottas, and Steven N MacEachern. 2005. “Bayesian Nonparametric Spatial Modeling With Dirichlet Process Mixing.” Journal of the American Statistical Association 100 (471): 1021–35.
Ishwaran, Hemant, and Lancelot F James. 2001. “Gibbs Sampling Methods for Stick-Breaking Priors.” Journal of the American Statistical Association 96 (453): 161–73.
MacEachern, Steven N. 1999. “Dependent Nonparametric Processes.” In ASA Proceedings of the Section on Bayesian Statistical Science, 1:50–55. Alexandria, Virginia. Virginia: American Statistical Association; 1999.
MacEachern, Steven N, Athanasios Kottas, and Alan E Gelfand. 2001. “Spatial Nonparametric Bayesian Models,” 6.
Moraffah, Bahman, and Antonia Papandreou-Suppappola. 2022. “Bayesian Nonparametric Modeling for Predicting Dynamic Dependencies in Multiple Object Tracking.” Sensors 22 (1): 388.
Nieto-Barajas, Luis E., Igor Prünster, and Stephen G. Walker. 2004. “Normalized Random Measures Driven by Increasing Additive Processes.” Annals of Statistics 32 (6): 2343–60.
Quintana, Fernando A., Peter Müller, Alejandro Jara, and Steven N. MacEachern. 2022. “The Dependent Dirichlet Process and Related Models.” Statistical Science 37 (1): 24–41.
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