Measure-valued stochastic processes

Including completely random measures and many generalizations

Often I need to have a nonparametric representation for a measure over some non-finite index set. We might want to represent a probability, or mass, or a rate. I might want this representation to be something flexible and low-assumption, like a Gaussian process. If I want a nonparametric representation of functions this is not hard; I can simply use a Gaussian process. What can I use for measures?

If I am working directly with random distributions of mass then I might want other properties like conservation of mass.

Completely random measures

See Kingman (1967) for the OG introduction. Foti et al. (2013) summarises

A completely random measure (CRM) is a distribution over measures on some measurable space $$\left(\Theta, \mathcal{F}_{\Theta}\right)$$, such that the masses $$\Gamma\left(A_{1}\right), \Gamma\left(A_{2}\right), \ldots$$ assigned to disjoint subsets $$A_{1}, A_{2}, \cdots \in \mathcal{F}_{\Theta}$$ by a random measure $$\Gamma$$ are independent. The class of completely random measures contains important distributions such as the beta process, the gamma process, the Poisson process and the stable subordinator.

AFAICT any subordinator will in fact do. A subordinator is an a.s. non-decreasing Lévy process.

TBC

Random coefficient polynomials

As seen in random spectral measures

A classic

Beta process

As seen, apparently, in survival analysis

Other measure priors

Various transforms of Gaussian processes seem popular, e.g. squared or exponentiated. These always seem too messy to me.

References

Barbour, A. D. n.d. Journal of Applied Probability 25 (A): 175–84.
Barbour, A.D., and T.C. Brown. 1992. Stochastic Processes and Their Applications 43 (1): 9–31.
Barndorff-Nielsen, O. E., and J. Schmiegel. 2004. Russian Mathematical Surveys 59 (1): 65.
Çinlar, E. 1979. Stochastic Processes and Their Applications 9 (2): 147–54.
Foti, Nicholas, Joseph Futoma, Daniel Rockmore, and Sinead Williamson. 2013. In Artificial Intelligence and Statistics, 20–28.
Griffiths, Thomas L., and Zoubin Ghahramani. 2011. Journal of Machine Learning Research 12 (32): 1185–1224.
Higdon, Dave. 2002. In Quantitative Methods for Current Environmental Issues, edited by Clive W. Anderson, Vic Barnett, Philip C. Chatwin, and Abdel H. El-Shaarawi, 37–56. London: Springer.
Hjort, Nils Lid. 1990. The Annals of Statistics 18 (3): 1259–94.
James, Lancelot F. 2005. Annals of Statistics 33 (4): 1771–99.
Kingman, John. 1967. Pacific Journal of Mathematics 21 (1): 59–78.
Kirch, Claudia, Matthew C. Edwards, Alexander Meier, and Renate Meyer. 2019. Bayesian Analysis 14 (4): 1037–73.
Lau, John W., and Edward Cripps. 2022. Bernoulli 28 (1): 638–62.
Lee, Juho, Xenia Miscouridou, and François Caron. 2019. arXiv:1905.10733 [Cs, Math, Stat], May.
Lijoi, Antonio, Bernardo Nipoti, and Igor Prünster. 2014. Bernoulli 20 (3): 1260–91.
Lijoi, Antonio, and Igor Prünster. 2010. In Bayesian Nonparametrics, edited by Nils Lid Hjort, Chris Holmes, Peter Müller, and Stephen G. Walker. Cambridge University Press.
Lin, Jiayu. 2016. “On The Dirichlet Distribution,” 75.
Liou, Jun-Jih, Yuan-Fong Su, Jie-Lun Chiang, and Ke-Sheng Cheng. 2011. Stochastic Environmental Research and Risk Assessment 25 (2): 235–51.
Lo, Albert Y., and Chung-Sing Weng. 1989. Annals of the Institute of Statistical Mathematics 41 (2): 227–45.
Max-K. von Renesse. 2005. Technische Universität Berlin.
Meier, Alexander. 2018.
Meier, Alexander, Claudia Kirch, and Renate Meyer. 2020. Journal of Multivariate Analysis 175 (January): 104560.
Nieto-Barajas, Luis E., Igor Prünster, and Stephen G. Walker. 2004. Annals of Statistics 32 (6): 2343–60.
Pandey, Gaurav, and Ambedkar Dukkipati. 2016. In International Conference on Machine Learning, 1605–13. PMLR.
Ranganath, Rajesh, and David M. Blei. 2018. Journal of the American Statistical Association 113 (521): 417–30.
Rao, Vinayak, and Yee Whye Teh. 2009. “Spatial Normalized Gamma Processes.” In Proceedings of the 22nd International Conference on Neural Information Processing Systems, 1554–62. NIPS’09. Red Hook, NY, USA: Curran Associates Inc.
Roychowdhury, Anirban, and Brian Kulis. 2015. In Artificial Intelligence and Statistics, 800–808. PMLR.
Thibaux, Romain, and Michael I. Jordan. 2007. In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics, 564–71. PMLR.
Walker, Stephen G., Paul Damien, PuruShottam W. Laud, and Adrian F. M. Smith. 1999. Journal of the Royal Statistical Society: Series B (Statistical Methodology) 61 (3): 485–527.
Wolpert, R., and Katja Ickstadt. 1998. Biometrika 85 (2): 251–67.
Wolpert, Robert L., and Katja Ickstadt. 1998. In Practical Nonparametric and Semiparametric Bayesian Statistics, edited by Dipak Dey, Peter Müller, and Debajyoti Sinha, 227–42. Lecture Notes in Statistics. New York, NY: Springer.

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