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Daan Wierstra coauthored the 2014 work on stochastic backpropagation with Danilo Jimenez Rezende and Shakir Mohamed. The primary paper identifies all three with Google DeepMind in London. Their method jointly trains a generative model and a recognition network representing an approximate posterior.^[ref]

An original scalar example isolates the shared inference-map parameter. For inputs $x\in \{0,1\}$, take a Gaussian sample $Z_w(x)=wx+ϵ$, where $ϵ$ has a standard normal distribution. The same $w$ controls both conditional means. Define the chosen toy lossIts derivative is $w-1$, so at $w=0$ the gradient is $-1$. Holding a noise draw fixed permits the chain rule to pass through $Z_w(x)$ to the shared parameter, rather than optimizing a separate free mean for every input.

The paper's actual objective is a variational bound, not this squared-error illustration. Its Gaussian reparameterization provides a computable reparameterization gradient through random variables.^[ref] The example explains one ingredient of that collaborative contribution without assigning the complete method to one coauthor.

Danilo Jimenez Rezende coauthored the 2014 stochastic-backpropagation work with Shakir Mohamed and Daan Wierstra, developing trainable generative and approximate-inference models.^[ref] With Shakir Mohamed, he subsequently developed normalizing flows for richer approximations in variational inference.^[ref]

An original one-dimensional transformation illustrates the density accounting. Let $ϵ$ have standard normal distribution and take $Z=2ϵ+1$. Sampling is straightforward, but its density must also account for the stretching of length: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.

Shakir Mohamed coauthored the 2014 stochastic-backpropagation paper with Danilo Jimenez Rezende and Daan Wierstra. It develops gradient computations through stochastic variables for joint learning of generative and approximate-inference models.^[ref] He also coauthored the 2015 work on normalizing flows with Danilo Jimenez Rezende, constructing flexible approximate posterior distributions through invertible transformations.^[ref]

An original smooth example shows why a reparameterization gradient is a computational object. Let $Z_{\mu }=\mu +ϵ$ with standard-normal $ϵ$, and choose $J(\mu )=\mathbb{E}[\sin Z_{\mu }]$. The exact expectation and derivative areDifferentiating a sampled value while holding its noise fixed gives $\cos (\mu +ϵ)$. At $\mu =0$, a chosen draw $ϵ=0$ yields gradient $1$, while the exact expectation gradient is $e^{-1/2}\approx 0.60653$. Averaging the sampled gradients over the stated noise distribution recovers the expectation gradient in this smooth bounded example.

That calculation depends on the distribution and differentiability assumptions. It does not make a single random derivative equal to the derivative of an expectation, nor does it by itself establish an analogous rule for a discontinuous sampling map. The cited contributions concern explicit probabilistic models and estimators, not an unrestricted license to differentiate through any random procedure.

Primary ICML2014 paper by Danilo Jimenez Rezende, Shakir Mohamed and Daan Wierstra, PMLR32(2), 1278–1286. Section3.2, equation(10), gives the Gaussian location-scale transformation and its pathwise gradient; sections2 and4 describe generative and recognition models.