Solved – Standard notation for [sampling from] a conditional probability

In most probability and statistics books, the notation$$Xsim fquadtext{or}quad Xsim f(x)$$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 $$varphi(x)=frac{1}{sqrt{2pi}}exp{-x^2/2}$$over $mathbb{R}$ and people often write $Xsimvarphi(x)$ 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 $$X|Y=ysim f(x|y)$$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 $U_t$.