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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 μ and/or (positive) multiplier σ, Stan uses an affine transform. Such a variable X is transformed to a new variable Y, where
Y=μ+σ∗X.
The default value for the offset μ is 0 and for the multiplier σ is 1 in case not both are specified.
Affine Inverse Transform
The inverse of this transform is
X=Y−μσ.
10.5.1 Absolute Derivative of the Affine Inverse Transform
The absolute derivative of the affine inverse transform is
|ddy(y−μσ)|=1σ.
Therefore, the density of the transformed variable Y is
pY(y)=pX(y−μσ)⋅1σ.