Drawing numbers using the CDF

Question

Say I have a (generally high-dimensional) random variable $X$ with known, continuous CDF $F(X)$.

Is there a good algorithm for drawing values of $X$ that doesn't require that I calculate the joint density?

I'm specifically interested in the GEV distribution used in the latent variable formulation of the nested logit: $$F\left(\vec\nu\right) = \exp\left(- ∑_{n ∈ N} \left(∑_{k∈n} \exp\left(\frac{-\nu_k}{λ_n}\right)\right)^{λ_n}\right).$$ That distribution should have analytic PDFs, but they're pretty gross (and are going to get even grosser with multiple layers of nests).

[Edit: this question was labelled a duplicate of Inverse of cumulative density function for Multivariate Normal Distribution. It's not obvious why -- is the assumption that the inverse-transform algorithm is the only algorithm for drawing from a CDF? That doesn't seem right.]

Answer 1

The MCMC-MH algorithm requires the ratio of the PDFs at the proposal value $x'$ and the existing value $x_t$. Let $K$ be the dimension of $X$. We can approximate the ratio by looking at the ratio of probabilities that $X$ is in a neighbourhood of $x'$ or $x_t$ $$ \frac{f(x')}{f(x)} = \frac{\lim_{e \to 0}P[X \in x' \pm e]/e^K}{ \lim_{e \to 0}P[X \in x_t \pm e]/e^K} = \lim_{e \to 0} \frac{P[X \in x' \pm e]}{ P[X \in x_t \pm e]} \approx \frac{P[X \in x' \pm \epsilon]}{ P[X \in x_t \pm \epsilon]}, $$ where the approximation should hold for small $\epsilon$.

However it's not obvious how to use that approximation in log-space, which will likely be important given that both the numerator and the denominator will be very close to 0. In particular, $$ \log P[X \in x' \pm \epsilon] = \log\left(P[X < x' + \epsilon] - P[X < x' - \epsilon] \right) \\ \neq \log\left(P[X < x' + \epsilon]\right) - \log \left(P[X < x' - \epsilon] \right). $$