# Why are density functions sometimes written with conditional notation?

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

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

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?

• It may simply be due to different people's idiosyncrasies. I would expect different authors to be more or less consistent about this. Oct 27 '20 at 23:27
• How do you verbalize this? "f of x given mu, sigma..."?
– smci
Oct 28 '20 at 19:09
• Related (if not duplicate): stats.stackexchange.com/questions/130869 see also stats.stackexchange.com/questions/156617 Oct 28 '20 at 19:46

• 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 $$L(\theta; x) := f(x \mid \theta)$$.
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.