 Gaussian reparameterization

Gaussian reparameterization represents a random variable with a normal distribution using a location, a scale and parameter-independent noise:Here $\epsilon$ has the standard normal distribution. The resulting $Z$ has expected value $\mu$ and variance ${\sigma }^2$. The probability distribution is unchanged; the representation separates the random draw from the adjustable parameters. This makes reparameterization gradients available for differentiable expectations, including the Gaussian approximate posterior in a variational autoencoder.^[ref]

For a complete example, fix $\sigma$ and considerHolding the same noise value fixed while differentiating givesAt $\mu =1$, $\sigma =2$, the particular noise draw $\epsilon =1/2$ gives $Z=2$ and derivative estimate $4$, while the true derivative is $2$. An individual sample need not equal the derivative; the equality is in expected value.

Parameterizing a positive scale as $\sigma =e^{\rho }$ also gives $\partial Z/\partial \rho =e^{\rho }\epsilon$. One can differentiate through the deterministic transformation while leaving the noise law fixed. For a general integrand, exchanging derivative and expectation still requires justified differentiation under the integral sign; the representation alone is not a proof of that exchange. Reusing noise when parameters change also connects the construction to common random numbers.