Standard notation for [sampling from] a conditional probability

Question

I am looking for the correct formal notation of sampling from a continuous conditional probability. In [1] (Table 4.3) they write:

$$\text{sample } x_t^{[m]} \sim p(x_t | u_t, x_{t-1}^{[m]})$$

Should the word 'sample' be used? And does $x_t$ in $p(x_t|..)$ per definition refer to the chosen $x_t^{[m]}$?

In my case I have some conditional probability $p(s|o)$, which is a probability function, for example Normal, and depends on $o$:

$$p(s|o) = \mathcal{N}(s;o,\sigma^2)$$

And then I want to sample from this distribution: $s \sim p(s|o)$. With this I want to say that sample $s$ is taken based on a normal distribution probability with $o$ as mean. Is this correct?

[1] Thrun, S., Burgard, W., & Fox, D. (2005). Probabilistic Robotics (Intelligent Robotics and Autonomous Agents. The MIT Press.

Answer 1

In most probability and statistics books, the notation$$X\sim f\quad\text{or}\quad X\sim f(x)$$means that the random variable is distributed from the probability distribution with density $f$ [with respect to an implicit dominating measure]. Most often, people use $f(x)$ to represent the function $f$, where $x$ is a dummy notation. For instance, the Gaussian density $\varphi$ is defined by $$\varphi(x)=\frac{1}{\sqrt{2\pi}}\exp\{-x^2/2\}$$over $\mathbb{R}$ and people often write $X\sim\varphi(x)$ where $x$ is merely a symbol that reminds us $f$ is a function.

The $|$ sign used in conditional probability notations was introduced by Harold Jeffreys. When writing $$X|Y=y\sim f(x|y)$$it means that the distribution of the random variable $X$ conditional on the realisation $y$ of the random variable $Y$ has a density $f(\cdot|y)$ or $f(x|y)$. In this notation, $x$ is a dummy that can take any value, while $y$ is the conditioning realisation of $Y$. In your example, $$x_t^{[m]} \sim p(x_t | u_t, x_{t-1}^{[m]})$$ means that the random variable $x_t^{[m]}$ [should be $X_t^{[m]}$ to distinguish the random variable from its realisation $x_t^{[m]}$] has the density $p(\cdot | u_t, x_{t-1}^{[m]})$ given the realisation $x_t^{[m]}$ of the random variable $X_t^{[m]}$ and the realisation $u_t$ of the random variable $U_t$.