Conceptual/Notational question on conditional distributions and "given"

Question

So in the book I'm reading, I see the notations $f(x|\theta)$ being used to refer to population distributions, dependent on $\theta$ which are in a family. The author explains this as a notational convenience to keep track of the parameter $\theta$. My question is, is this the same as a conditional pdf? Can I write things like $$f(x|\theta)=f(x,\theta)/(f(\theta))$$. Also, what is the difference between $f(x|\theta)$ and just the marginal pdf of x, $f(x)$? I don't see the purpose of constantly conditioning on the parameter. This theme is continued for expectations as well. I know the distribution is dependent on its parameters, but why do we need to analyze the conditional distribution? Why would we take $\theta$ as given if its the very thing we are trying to find out about? Thanks for any help.

Answer 1

By "notational convenience" the author really means "abuse of notation". It is not conditioning because $\theta$ is not random, and the display you wrote doesn't make sense for that reason (what would $f(\theta)$ be)? Many authors prefer something like $f_\theta(x), E_\theta(x)$ etc. What the author means by $f(x|\theta)$ is in fact the marginal density $f(x)$, using $\theta$ to index a family of distributions. Eg, you might write $f(x|\mu)=\phi(x-\mu)$ to refer to a normal density with mean $\mu$. Here the family is normal distributions with any mean, and $f(x|\theta),f_\theta(x),f(x|\mu),f_\mu(x)$ etc picks out one such normal distribution. This response should all be qualified by saying that in a bayesian framework, where $\theta$ is in fact a random variable, $f(x|\theta)$ would not be an abuse of notation, and bayes rule would apply.