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.
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 , such that the masses assigned to disjoint subsets by a random measure 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
Random coefficient polynomials
As seen in random spectral measures. TBC
Other
Various transforms of Gaussian processes seem popular, e.g. squared or exponentiated. These always seem too messy to me.
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