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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

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

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

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

10.5.1 Absolute Derivative of the Affine Inverse Transform

The absolute derivative of the affine inverse transform is

$\left| \frac{d}{dy} \left( \frac{y-\mu}{\sigma} \right) \right| = \frac{1}{\sigma}.$

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

$p_Y(y) = p_X \! \left( \frac{y-\mu}{\sigma} \right) \cdot \frac{1}{\sigma}.$