1
$\begingroup$

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,

  • 2
    Well, $f(x)$ is not a really good name. In principle, there is no confusion with the original $f(x,y)$, since that is a two variable function. But if you integrated $x$ out, and called the resulting function $f(y)$, there *would* be confusion.2012-09-06

2 Answers 2

0

You can use F(x) = ∫f(x,y)dy, and I studied Marginal density functions in that way only.

  • 0
    Not the best choice if there is any possibility of the other marginal $\int f(x,y)\,dx$ entering the picture. You can't call the other one $F$, but if you use another letter like $G$, there is an apparent asymmetry between $(F,G)$ since only one is an uppercase form of the original integrand.2012-09-09
0

Besides probability, marginals come up in the Kantorovich formulation of the optimal transportation problem. Wikipedia article uses the letters $\mu$ and $\nu$ for the marginals, which is consistent with other sources I've seen. You may consider adapting it. In any case I suggest using letters other than $f$.