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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.

Durk Kingma is a computer scientist who publishes as Diederik P. Kingma. His personal profile describes research on machine learning and generative models, and records a 2017 doctorate at the University of Amsterdam.^[ref] The institutional thesis front matter gives his full formal name as Diederik Pieter Kingma.^[ref]

Kingma and Max Welling coauthored Auto-Encoding Variational Bayes by Diederik P. Kingma and Max Welling. A variational autoencoder couples a latent-variable model with an approximating posterior distribution that can be optimized through variational inference. Their reparameterization gradients represent a simulated random variable as a differentiable transformation of parameter-independent noise, allowing derivatives through the transformation.^[ref] Kingma also coauthored the Adam paper with Jimmy Lei Ba.^[ref]

An original scalar calculation makes the noise handling concrete. Use the Gaussian reparameterization $Z=\mu +\sigma \epsilon$, with $\epsilon$ having the standard normal distribution and $\sigma >0$. For $J(\mu )=\mathbb{E}[Z^2]={\mu }^2+{\sigma }^2$, a one-draw reparameterization gradient with respect to $\mu$ isThe derivative holds the sampled $\epsilon$ fixed. This pathwise gradient is justified here by direct calculation; a general interchange of derivative and expected value requires differentiation under the integral sign conditions.

Noise coupling matters even for a central finite difference. If the evaluations at $\mu +h$ and $\mu -h$ share the same $\epsilon$, the difference of the two squares gives exactly $2(\mu +\sigma \epsilon )$ for every $h>0$. This is a use of common random numbers. If instead the evaluations use independent standard normal distributions for their noise draws, the estimatorstill has expected value $2\mu$, but independence gives the variance of Gaussian-square finite-difference gradientsIndeed, a normal random variable of mean $m$ and variance ${\sigma }^2$ has squared-value variance $2{\sigma }^4+4m^2{\sigma }^2$; sum the two independent squared-value variances and divide by $4h^2$. Thus shrinking $h$ need not suppress simulation noise.

For chosen $\mu =1$, $\sigma =2$ and $h=0.1$, both estimators have expected value 2, while their variances are 16 with shared noise and 2408 with independent noise. The example is a separate exact quadratic model; its purpose is to explain the role of keeping base noise fixed, not to reproduce a published variational-autoencoder benchmark.

A recognition network is a learned mapping from observed data to parameters of an approximate posterior distribution over latent variables. In a variational autoencoder, it specifies $q_{\varphi }(z\mid x)$, while the generative decoder specifies the opposite conditional direction $p_{\theta }(x\mid z)$. The recognition network supports amortized inference; it need not coincide with the exact posterior of the learned generative model.^[ref]

For a scalar Gaussian recognition law, the neural network can output a mean ${\mu }_{\varphi }(x)$ and log variance ${\ell }_{\varphi }(x)$. Thenwhere $\epsilon \sim N(0,1)$ is parameter-independent noise. Exponentiating the log variance ensures a positive variance, and the factor $1/2$ converts it to the standard deviation used in Gaussian reparameterization.^[ref]

For an original calculation, suppose one input gives $\mu =1$ and $\ell =\log 4$. The recognition variance is4 and its standard deviation is2. A noise draw $\epsilon =1/2$ gives $z=2$, while the posterior mean supplied by the same network remains1. Confusing log variance with log standard deviation would instead produce the wrong scale.

A deterministic network evaluation and a random latent sample are distinct steps. Re-evaluating the same fixed network on the same input returns the same distribution parameters; new noise draws can still yield different $z$ values. Returning only the mean omits the uncertainty represented by that distribution. Its adequacy must be assessed against the intended inference problem, not inferred merely from the fact that it outputs a positive variance.

A variational autoencoder is a generative model with a latent variable $z$ and a learned approximation to its posterior distribution given an observation $x$. Write the model as $p_{\theta }(x,z)=p(z)p_{\theta }(x\mid z)$. A recognition network, implemented as a neural network, produces the parameters of $q_{\varphi }(z\mid x)$; a generative decoder specifies $p_{\theta }(x\mid z)$. Using one recognition rule across observations is amortized inference, rather than solving a fresh independent optimization problem for every input.^[ref]

Training maximizes an evidence lower bound, often writtenThe first term scores reconstruction under the declared likelihood; the second compares the inferred latent law with the prior. The identityproves that it is a lower bound, since Kullback-Leibler divergence is nonnegative. It becomes tight when the recognition law equals the model posterior, under the usual density/support conditions.^[ref]

For a one-dimensional Gaussian example, let the prior be $N(0,1)$ and the recognition law be $N(\mu ,{\sigma }^2)$. Substituting their log densities and using $\mathbb{E}[z^2]={\mu }^2+{\sigma }^2$ givesAt $\mu =1$, $\sigma =2$, this is $2-\log 2\simeq 1.30685$. The value is a penalty for this particular latent law, not the whole training loss. Gaussian reparameterization writes $z=\mu +\sigma \epsilon$ with fixed-law standard-normal noise so that gradients can pass through the sampled latent value. After training, generation instead draws from the prior and then the generative decoder; it need not start from an observed input.