Measure-valued random variates

Completely random measures
Dirichlet processes
Using Gamma processes
Random coefficient polynomials
For categorical variables
Pitman-Yor
Indian Buffet process
Beta process
Other
Dependent
Incoming
References

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.

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

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.

Using Gamma processes

Random coefficient polynomials

As seen in random spectral measures. TBC

For categorical variables

A classic.

Pitman-Yor

Indian Buffet process

Beta process

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

Other

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

Dependent

See measure-valued processes.

Incoming

Lectures on the Poisson Process

References

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Çinlar, E. 1979. “On Increasing Continuous Processes.” Stochastic Processes and Their Applications 9 (2): 147–54.

Foti, Nicholas, Joseph Futoma, Daniel Rockmore, and Sinead Williamson. 2013. “A Unifying Representation for a Class of Dependent Random Measures.” In Artificial Intelligence and Statistics, 20–28.

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Griffiths, Thomas L., and Zoubin Ghahramani. 2011. “The Indian Buffet Process: An Introduction and Review.” Journal of Machine Learning Research 12 (32): 1185–1224.

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