# How to interpret this notation with a tilde?

The following convention is sometimes encountered in the text (here example from HMM statement, $\{X_n\}_{n \geq 1}$ is a Markov process):

$$X_n \mid (X_{n-1} = x_{n-1}) \sim f(x_n \mid x_{n-1})$$

It is clear, what it means for the "plain" case (Why are probability distributions denoted with a tilde?), but what about the meaning of full formal notation when we have the conditional part $\mid\ (X_{n-1} = x_{n-1})$?

• It's describing the conditional distribution (rather than the unconditional distribution). Dec 17, 2015 at 9:05

The distribution of $X_n$, conditional on $X_{n-1}$ taking the value $x_{n-1}$, is described by the probability density function $f(x_n \mid x_{n-1})$.
• Thanks, could you please also write it in another notation, as I believe the original is just an abbreviation, or is the $X|Y$ the most used one? (If my guess is right, it should be something with p()) Dec 17, 2015 at 9:03
• $X\mid Y$ is standard. For example, $P( X=x \mid Y =y)$, often written $P(X\mid Y)$ is the probability that $X$ takes the value $x$ given $Y$ takes the value $y$. $E[X \mid Y=y]$ is the expectation of $X$ conditional on $Y$ takes value $y$. $X \mid Y$ in general refers to $X$ conditional on $Y$. Dec 17, 2015 at 9:07
• Even better would be to translate it to English as: "The distribution of $X_n$, conditional on $X_{n-1}$ taking the value $x_{n-1}$, is described by the probability density function $f(\ \mid x_{n-1})$."