= Danilo Jimenez Rezende
{c}
{tag=Mathematics author}
<Danilo Jimenez Rezende> coauthored the 2014 stochastic-backpropagation work with <Shakir Mohamed> and <Daan Wierstra>, developing trainable generative and approximate-inference models.<Stochastic Backpropagation and Approximate Inference in Deep Generative Models by Danilo Jimenez Rezende and others>[Primary full byline; §§2–4]{ref} With <Shakir Mohamed>, he subsequently developed <normalizing flows> for richer approximations in <variational inference>.<Variational Inference with Normalizing Flows by Danilo Jimenez Rezende and Shakir Mohamed>[Primary PDF full byline; §3, change-of-variables construction]{ref}
An original one-dimensional transformation illustrates the density accounting. Let $\epsilon$ have standard <normal distribution> and take $Z=2\epsilon+1$. Sampling is straightforward, but its density must also account for the stretching of length:
$$
p_Z(z)=\frac12p_\epsilon\!\left(\frac{z-1}{2}\right).
$$
At $z=1$, this is $1/(2\sqrt{2\pi})$. The factor $1/2$ comes from the absolute derivative of the inverse <affine map>. Omitting it would give a density integrating to $2$ instead of $1$.
The example is only an affine Gaussian transformation; it does not claim that every nonlinear transformation preserves a Gaussian family. A <normalizing flow> composes suitable invertible transformations while tracking the corresponding density changes. This connects an efficient sampling procedure to the density evaluation needed in a variational objective, rather than treating the generated samples alone as a complete inference method.
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