I'm reading Peter Orbanz's notes on Bayesian nonparametrics http://stat.columbia.edu/~porbanz/papers/porbanz_BNP_draft.pdf. In it, he uses the following notation, which isn't defined.
We have some Dirac measure $\delta_{\phi_{k}}$, the atoms of a probability (mixing) measure \begin{align} \theta(\cdot)&=\sum_{k\in \mathbb{N}}c_{k}\delta_{\phi_{k}}(\cdot) \end{align} with atom locations $\phi_{k}$.
We also have \begin{align} \Theta&=\sum_{k\in \mathbb{N}}C_{k}\delta_{\Phi_{k}} \end{align}
We want to calculate \begin{align} P(\Phi_{i}\in d\phi|...)\\ P(\phi_{i}\in \cdot) \end{align}
I don't understand the following in terms of notation
- What does $\in d\phi$ mean? Particularly, what does $d\phi$ mean in this setting? I also see a distribution $G$, and they write $G(d\phi)$, which seems to indicate that $G$ is parametrized by $d\phi$, but I'm not sure.
- What does $\cdot$ mean as used in $\theta(\cdot)$ and $\in \cdot$ mean?