Let $f$ be a function $f : x \mapsto y$ where $x \in \mathbb{R}^n$ and $y \in \mathbb{R}^m$; $m \geq n$. $f$ is not invertible. I have a random variable $X$ s.t. $X \sim \mathcal{N}(\mu, \Sigma)$ where mean $\mu \in \mathbb{R}^n$ and covariance matrix $\Sigma \in \mathbb{R}^{n \times n}$. If it makes things easier, $\Sigma$ can be considered to be diagonal.
Is there an analytical way to calculate the probability (or log-likelihood) of $f(X)$? What about its gradients with respect to $\mu$ and $\sigma$? If not, is there a good way to estimate these?
EDIT: I'd be curious about the case where $f$ is invertible as well, as I may be able to tweak my problem, but it is suboptimal.