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: $$\hat\mu,\hat\sigma|X= \underset{\mu,\sigma}{\operatorname{argmax}} \sum_i \ln f(x_i\in X|\mu,\sigma) $$

So, this notation emphasizes that you use the PDF of the data set conditional on a candidate set of parameters to obtain the likelihood. Then you pick the set that maximizes the likelihood as your solution. Thus, the solution is truly conditional on the data set. That's why this notation is good for didactic purpose to show how the conditions sort of "flip" on left- and right-hand side.

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.

This notation is used often in MLE context to differentiate it from likelihood function and the estimation of the parameters conditional on data.

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: $$\hat\mu,\hat\sigma|X= \underset{\mu,\sigma}{\operatorname{argmax}} \sum_i \ln f(x_i\in X|\mu,\sigma) $$

So, this notation emphasizes that you use the PDF of the data set conditional on a candidate set of parameters to obtain the likelihood. Then you pick the set that maximizes the likelihood as your solution. Thus, the solution is truly conditional on the data set. That's why this notation is good for didactic purpose to show how the conditions sort of "flip" on left- and right-hand side.