In probability integrating out a variable is viewed as marginalization; One probability function turns into another probability function. In other cases and fields, taking a regular function as example, for example $f(x,y)$, when I integrate out $y$,
$ \int_{-\infty}^{\infty}f(x,y)dy $
which would give me a function based on only $x$. My question: question is when would this particular way of summarizing necessary (other than probability)? What are the common places this is used? Another one: can I call this function $f(x)$? It seems like the answer is no because I am not realy getting "the value of $f$ for $x$"; I believe I get a different function, and I lose some information,and lose it in a particular way.
Thanks,