What does it mean to integrate over a random measure?

I’m currently looking at a paper of Dirichlet process random effects model and the model specification is as follows:
yi=Xiβ+ψi+ϵiψiGGDP(α,G0)
where α is the scale parameter and G0 is the base measure. Later on in the paper, it suggests that we integrate a function over the base measure G0 such as
f(yj|θ,ψj)dG0(ψj). Is the base measure in Dirichlet process a c.d.f. or is it a p.d.f.? What happens if the base measure is a Gaussian?

Answer

Denote by M a measurable space of probability measures, containing the realisations of the Dirichlet process. The random probability measure G is a measurable function
G:ωGωM
and the integral with respect to G is the random variable
f(|ψ)dG(ψ):ωf(|ψ)dGω(ψ).
Thus f(|ψ)dG(ψ) is itself a random p.d.f. (if f(|ψ) is a p.d.f.).


The idea is that ψi follows some unknown distribution G. In some cases, you may have reasons to believe that ψi is normally distributed and then put a prior on the mean and variance. In other cases, you don’t want to make such parametric assumptions. In your model, for instance, the prior on G is a Dirichlet process.


Is the base measure in Dirichlet process a c.d.f. or is it a p.d.f.?

The base measure is any probability measure, usually taken to have full support. In some cases, it can be represented by a probability density function. This is not very important.

Attribution
Source : Link , Question Author : Daeyoung Lim , Answer Author : Olivier

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