Change of Variable Theorem

Suppose we have some random variable \(z_0 \in \mathbb{R}^n\) with distribution \(p(z_0)\) and we are given a differentiable function \(f: \mathbb{R}^n \rightarrow \mathbb{R}^n\). Define

\[z_1 = f(z_0)\]

What is the distribution of \(p(z_1)\)? We can see that

\[\partial z_1 = \partial f(z_0) = \partial_{z_0} f * \partial z_0\]

Consequently, if we take the log probability, we have:

\[\log p(z_1) = \log p(z_0) + \log \det(\frac{\partial f}{\partial z_0})\]