# Measure-valued random variates

Including completely random measures and many generalizations

October 16, 2020 — March 30, 2022

Often I need a nonparametric representation for a measure over some non-finite index set. We might want to represent a probability, 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 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 do, i.e. any a.s. non-decreasing Lévy process.

TBC

## 2 Dirichlet processes

Random locations plus random weights give 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

## 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

## 11 Incoming

## 12 References

*Journal of Applied Probability*.

*Stochastic Processes and Their Applications*.

*Russian Mathematical Surveys*.

*Stochastic Processes and Their Applications*.

*Artificial Intelligence and Statistics*.

*Electronic Journal of Statistics*.

*Journal of Machine Learning Research*.

*Quantitative Methods for Current Environmental Issues*.

*The Annals of Statistics*.

*Journal of the American Statistical Association*.

*Annals of Statistics*.

*Pacific Journal of Mathematics*.

*Bayesian Analysis*.

*Bernoulli*.

*arXiv:1905.10733 [Cs, Math, Stat]*.

*Bernoulli*.

*Bayesian Nonparametrics*.

*Stochastic Environmental Research and Risk Assessment*.

*Annals of the Institute of Statistical Mathematics*.

*Communications for Statistical Applications and Methods*.

*Journal of Multivariate Analysis*.

*Annals of Statistics*.

*International Conference on Machine Learning*.

*Proceedings of the 22nd International Conference on Neural Information Processing Systems*. NIPS’09.

*Artificial Intelligence and Statistics*.

*Proceedings of the Eleventh International Conference on Artificial Intelligence and Statistics*.

*Journal of the Royal Statistical Society: Series B (Statistical Methodology)*.

*Practical Nonparametric and Semiparametric Bayesian Statistics*. Lecture Notes in Statistics.

*Biometrika*.

*ACM Computing Surveys*.