Let $X$ and $Y$ in $\Bbb{R}^n$ be two random vectors. We assume that $X\mid Y\sim\mathcal{N}(Y,\Sigma_X)$ and $Y\sim\mathcal{N}(\mu_Y,\Sigma_Y)$
The goal is to sample from the distribution of $X$.
If we are given $Y$ then we can use $p_{X\mid Y}(x\mid y)$, which is known.
If we are given some $x\in X$, then how can we do this sampling? I think that we need the marginal density $p_X(x)$, which should be given (using Bayes' rule) by
$$ p_X(x) = \frac{p_Y(y)}{p_{Y\mid X}(y\mid x)}p_{X\mid Y}(x\mid y). $$
If I am not mistaken, we need to find the ratio $\frac{p_Y(y)}{p_{Y\mid X}(y\mid x)}$.