What does $\in$ mean vs $=$ in probability? What does $d\phi$ mean?

Question

In the following lecture notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf, I often see something like \begin{align} P[\Phi_{i}\in d\phi|...]\\ P[\Phi_{i}=d\phi|...] \end{align} (see for instance pages 14 and 15 of Peter Orbanz' notes).

Where $\Phi_{k}$ are the atom locations of a random mixing measure, and $\phi$ is some value that $\Phi_{k}$ can take. What is the difference between $\in$ and $=$?

Also, what does the $d\phi$ mean here? I don't really understand the intuition. Particularly, if $G$ is some distribution, what does $G(d\phi)$ mean? I would think it would mean that $G$ is parametrized by whatever is in parentheses, but I don't think that $d\phi$ are parameters.

Answer 1

This is indeed rather confusing: the notation $d\phi$ stands for an infinitesimal measurable set located around $\phi$. As in standard measure theory settings with Leibniz's $dx$. It can thus be used in integrals as $$\mathbb{P}(\Phi_k^*\in A|B_{1:n},X_{1:n})=\int_A \mathbb{P}(\Phi_k^*\in d\phi|B_{1:n},X_{1:n})$$ to borrow from eqn (2.33) in Peter Orbanz' notes [p.14]. Because of the atomic nature of the random variable $\Phi_i$ given the cluster parameters $\Phi_k^*$, you can also use the notation $$\mathbb{P}(\Phi_i=d\phi|B_{-i},\Phi_k^*,X_{i}=x_i)$$ as in eqn (2.34).

To answer completely your question,

1. The $\Phi_k^*$'s are the atom locations, not the $\Phi_i$'s;
2. $G$ is the functional parameter or reference measure of the Dirichlet process, which provides the generated $\Phi^*_k$'s in the infinite mixture representation; $\phi$ is thus a dummy value of those $\Phi^*_k$'s, just like the density notation $f(x)$ involves a dummy $x$. Thus, $\phi$ is not a parameter of $G$;