Sample from conditional distribution knowing the joint distribution

Question

I know, if you plug-in specific values for the conditioned variables into the joint density function, you do not get the same (density) value as if you plug in the same values into the conditional distribution, but I assume they are proportional, such that sampling from the joint given values for the conditioned variables, is the same as sampling from the conditional distibution directly.

Same, in the sense that resulting sequences will have the same distribution (the conditional distribution).

Am I correct?

This question is also implicit in this question

Answer 1

If you are given a joint density on a pair of random variables, say $$\pi(x,y)$$ then for a given value of $x$, $x_0$ say, the conditional density of $Y$ given $X=x_0$ is a.s. equal to$$\pi(x_0,y)$$up to a multiplicative constant that is a function of $x_0$. It is therefore possible to sample from the conditional distribution when the joint density is available as a computable function.