I am coming from machine learning community, during reading papers, different authors use different notations for parameterized probability distribution and expectation. Thus, I am quite curious to see what is most preferable or acceptable notation to recommend from mathematics community.
There are a few examples as following
A probability density function over $x$
(a): $p(x)$
(b): $P(x)$
(c): $\mathbb{P}(x)$
(d): $\mathrm{P}(x)$
For following questions, by default using $P(x)$.
Given a probability distribution $P(x)$ parameterized by a vector $\theta$.
(a): $P(x|\theta)$, I don't like this, because it confuses with conditional probability distribution.
(b): $P(x; \theta)$, also not a big fun of it,
(c): $P_{\theta}(x)$, personal favorite
For following question, by default, using $P_\theta(x)$
Given a random variable $x$ with PDF $P(x)$ parameterized by $\theta$, the expectation operation
(a): $E[x|\theta]$ or $\mathbb{E}[x|\theta]$
(b): $E_\theta[x]$ or $\mathbb{E}_\theta[x]$
(c): $E_{x\sim P_\theta(x)}[x]$ or $\mathbb{E}_{x\sim P_\theta(x)}[x]$
(d): $E_{x\sim P_\theta(\cdot)}[x]$ or $\mathbb{E}_{x\sim P_\theta(\cdot)}[x]$