Measure-valued random variates

Including completely random measures and many generalizations

October 16, 2020 — March 30, 2022

Figure 1

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 (e.g. probability) mass then I might want conservation of mass, for example.

Processes are that naturally represent mass and measure are a whole field in themselves. Giving a taxonomy is not easy, but the same ingredients and tools tend to recur; Here is a list of pieces that we can plug together to create a random measure.

1 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 do, i.e. any a.s. non-decreasing Lévy process.

TBC

2 Dirichlet processes

Random locations plus random weights gives us a Dirichlet process. Breaking sticks, or estimation of probability distributions using the Dirichlet process. I should work out how to sample from the posterior of these. Presumably the Gibbs sampler from Ishwaran and James (2001) is the main trick.

3 Using Gamma processes

4 Random coefficient polynomials

As seen in random spectral measures. TBC

5 For categorical variables

A classic.

6 Pitman-Yor

7 Indian Buffet process

8 Beta process

As seen, apparently, in survival analysis (Hjort 1990; Thibaux and Jordan 2007).

9 Other

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

10 Dependent

See measure-valued processes.

11 Incoming

12 References

Barbour, A. D. n.d. Stein’s Method and Poisson Process Convergence.” Journal of Applied Probability.
Barbour, A.D., and Brown. 1992. Stein’s Method and Point Process Approximation.” Stochastic Processes and Their Applications.
Barndorff-Nielsen, and Schmiegel. 2004. Lévy-Based Spatial-Temporal Modelling, with Applications to Turbulence.” Russian Mathematical Surveys.
Çinlar. 1979. On Increasing Continuous Processes.” Stochastic Processes and Their Applications.
Foti, Futoma, Rockmore, et al. 2013. A Unifying Representation for a Class of Dependent Random Measures.” In Artificial Intelligence and Statistics.
Gil–Leyva, Mena, and Nicoleris. 2020. Beta-Binomial Stick-Breaking Non-Parametric Prior.” Electronic Journal of Statistics.
Griffiths, and Ghahramani. 2011. The Indian Buffet Process: An Introduction and Review.” Journal of Machine Learning Research.
Higdon. 2002. Space and Space-Time Modeling Using Process Convolutions.” In Quantitative Methods for Current Environmental Issues.
Hjort. 1990. Nonparametric Bayes Estimators Based on Beta Processes in Models for Life History Data.” The Annals of Statistics.
Ishwaran, and James. 2001. Gibbs Sampling Methods for Stick-Breaking Priors.” Journal of the American Statistical Association.
James. 2005. Bayesian Poisson Process Partition Calculus with an Application to Bayesian Lévy Moving Averages.” Annals of Statistics.
Kingman. 1967. Completely Random Measures.” Pacific Journal of Mathematics.
Kirch, Edwards, Meier, et al. 2019. Beyond Whittle: Nonparametric Correction of a Parametric Likelihood with a Focus on Bayesian Time Series Analysis.” Bayesian Analysis.
Lau, and Cripps. 2022. Thinned Completely Random Measures with Applications in Competing Risks Models.” Bernoulli.
Lee, Miscouridou, and Caron. 2019. A Unified Construction for Series Representations and Finite Approximations of Completely Random Measures.” arXiv:1905.10733 [Cs, Math, Stat].
Lijoi, Nipoti, and Prünster. 2014. Bayesian Inference with Dependent Normalized Completely Random Measures.” Bernoulli.
Lijoi, and Prünster. 2010. Models Beyond the Dirichlet Process.” In Bayesian Nonparametrics.
Lin. 2016. “On The Dirichlet Distribution.”
Liou, Su, Chiang, et al. 2011. Gamma Random Field Simulation by a Covariance Matrix Transformation Method.” Stochastic Environmental Research and Risk Assessment.
Lo, and Weng. 1989. On a Class of Bayesian Nonparametric Estimates: II. Hazard Rate Estimates.” Annals of the Institute of Statistical Mathematics.
MacEachern. 2016. Nonparametric Bayesian Methods: A Gentle Introduction and Overview.” Communications for Statistical Applications and Methods.
Meier. 2018. A matrix Gamma process and applications to Bayesian analysis of multivariate time series.”
Meier, Kirch, and Meyer. 2020. Bayesian Nonparametric Analysis of Multivariate Time Series: A Matrix Gamma Process Approach.” Journal of Multivariate Analysis.
Nieto-Barajas, Prünster, and Walker. 2004. Normalized Random Measures Driven by Increasing Additive Processes.” Annals of Statistics.
Paisley, Zaas, Woods, et al. n.d. “A Stick-Breaking Construction of the Beta Process.”
Pandey, and Dukkipati. 2016. On Collapsed Representation of Hierarchical Completely Random Measures.” In International Conference on Machine Learning.
Ranganath, and Blei. 2018. Correlated Random Measures.” Journal of the American Statistical Association.
Rao, and Teh. 2009. “Spatial Normalized Gamma Processes.” In Proceedings of the 22nd International Conference on Neural Information Processing Systems. NIPS’09.
Roychowdhury, and Kulis. 2015. Gamma Processes, Stick-Breaking, and Variational Inference.” In Artificial Intelligence and Statistics.
Thibaux, and Jordan. 2007. Hierarchical Beta Processes and the Indian Buffet Process.” In Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics.
von Renesse. 2005. Two Remarks on Completely Random Priors.”
Walker, Damien, Laud, et al. 1999. Bayesian Nonparametric Inference for Random Distributions and Related Functions.” Journal of the Royal Statistical Society: Series B (Statistical Methodology).
Wolpert, Robert L., and Ickstadt. 1998. Simulation of Lévy Random Fields.” In Practical Nonparametric and Semiparametric Bayesian Statistics. Lecture Notes in Statistics.
Wolpert, R., and Ickstadt. 1998. Poisson/Gamma Random Field Models for Spatial Statistics.” Biometrika.
Xuan, Lu, and Zhang. 2020. A Survey on Bayesian Nonparametric Learning.” ACM Computing Surveys.