Stochastic processes which represent measures over the reals

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.


A subordinator is an a.s. non-decreasing Lévy process. These can be used a random measures, which means they can be used as priors, which means they are handy for Bayesian nonparametrics. These priors include as a special case Dirichlet process priors in a way I will make precise maybe one day.

A rule of thumb is that while Gaussian process regression is good for giving you curves and more generally, fields, a subordinator prior can be interpreted as giving you non-negative measures. They are always kinda lumpy measures. It is easy to conserve mass with these things.

Other measure priors

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


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