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I keep seeing density functions that don't explicitly arise from conditioning written with the conditional sign: For example for the density of the Gaussian $N(\mu,\sigma)$ why write: $$ f(x| \mu, \sigma)=\frac{1}{\sqrt{2\pi \sigma^2}}\exp{-\frac{(x-\mu)^2}{2\sigma^2}}$$

instead of

$$ f(x)=\frac{1}{\sqrt{2\pi \sigma^2}}\exp{-\frac{(x-\mu)^2}{2\sigma^2}}$$

Is this done purely to be explicit as to what the parameter values are or(what I'm hoping for) is there some meaning related to conditional probability?

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  • In a Bayesian context, the parameters are random variables, so in that context the density is actually the conditional density of $X \mid (\mu, \sigma)$. In that setting, the notation is very natural.
  • Outside of a Bayesian context, it is just a way to make it clear that the density depends (here I am using this word colloquially, not probabilistically) on the parameters. Some people use $f_{\mu, \sigma}(x)$ or $f(x; \mu, \sigma)$ to the same effect.
  • This latter point can be important in the context of likelihood functions. A likelihood function is a function of the parameters $\theta$, given some data $x$. The likelihood is sometimes written as $L(\theta \mid x)$ or $L(\theta ; x)$, or sometimes as $L(\theta)$ when the data $x$ is understood to be given. What is confusing is that in the case of a continuous distribution, the likelihood function is defined as the value of the density corresponding to the parameter $\theta$, evaluated at the data $x$, i.e. $L(\theta; x) := f_\theta(x)$. Writing $L(\theta; x) = f(x)$ would be confusing, since the left-hand side is a function of $\theta$, while the right-hand side ostensibly does not appear to depend on $\theta$. While I prefer writing $L(\theta; x) := f_\theta(x)$, some might write $L(\theta; x) := f(x \mid \theta)$.
  • I have not really seen much consistency in notation across different authors, although someone more well-read than I can correct me if I am wrong.
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  • $\begingroup$ if you find point 3 confusing, that simply means that you don't understand likelihood; that kind of inversion is the whole point of having a likelihood function $\endgroup$
    – carlo
    Oct 28 '20 at 19:21
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This notation is used often in MLE context to differentiate it from likelihood function and the estimation of the parameters conditional on data.

In MLE you do something like this: $$\hat\mu,\hat\sigma|X= \underset{\mu,\sigma}{\operatorname{argmax}} \mathcal L(X|\mu,\sigma)$$ $$\mathcal L(X|\mu,\sigma)=\prod_i f(x_i\in X|\mu,\sigma) $$

So, this notation emphasizes that you use the PDF $f(.)$ of the data set conditional on a candidate set of parameters to obtain the likelihood function $\mathcal L$. Then you pick the set that maximizes the likelihood as your solution $\hat\mu,\hat\sigma$. Thus, the solution is truly conditional on the data set $X$, while the likelihood is conditional on the candidate parameter set $\mu,\sigma$. That's why this notation is good for didactic purpose to show how the conditions sort of "flip" on left- and right-hand side.

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