Gibbs Sampling - Calculating the full conditionals from the joint density

Question

Given a joint density, $f(x_1, x_2)$, can its pmf/pdf be found generally by the method outlined below:

For a joint density, $f(x_1, x_2)$ if we hold $x_2$ constant in the joint density, we will get the conditional density for $f(x_1 | x_2)$?

Here is the text from the textbook:

I am assuming the validity of it is due to the fact that $f(x_1, x_2) = f(x_1 | x_2) f(x_2)$ and that because the conditional density $f(x_1 | x_2)$ is purely a function of $x_1$?

Thanks for your help

Answer 1

Indeed, $f(x_1|x_2)\propto f(x_1,x_2)$ when the proportionality sign $\propto$ means that both sides are proportional as functions of $x_1$. The missing "constant" in the proportionality relation is however a function of $x_2$.