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10.5 Affinely transformed scalar

Stan uses an affine transform to be able to specify parameters with a given offset and multiplier.

Affine transform

For variables with expected offset $\mu$ and/or (positive) multiplier $\sigma$, Stan uses an affine transform. Such a variable $X$ is transformed to a new variable $Y$, where

$$Y = \frac{X - \mu}{\sigma}.$$

The default value for the offset $\mu$ is $0$ and for the multiplier $\sigma$ is $1$ in case not both are specified.

Affine inverse transform

The inverse of this transform is

$$X = \mu + \sigma \cdot Y.$$

10.5.1 Absolute derivative of the affine inverse transform

The absolute derivative of the affine inverse transform is

$$\left| \frac{d}{dy} \left( \mu + \sigma \cdot y \right) \right| = \sigma.$$

Therefore, the density of the transformed variable $Y$ is

$$p_Y(y) = p_X \! \left( \mu + \sigma \cdot y \right) \cdot \sigma.$$