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10.2 Lower Bounded Scalar
Stan uses a logarithmic transform for lower and upper bounds.
Lower Bound Transform
If a variable X is declared to have lower bound a, it is transformed to an unbounded variable Y, where
Y=log(X−a).
Lower Bound Inverse Transform
The inverse of the lower-bound transform maps an unbounded variable Y to a variable X that is bounded below by a by
X=exp(Y)+a.
Absolute Derivative of the Lower Bound Inverse Transform
The absolute derivative of the inverse transform is
|ddy(exp(y)+a)|=exp(y).
Therefore, given the density pX of X, the density of Y is
pY(y)=pX(exp(y)+a)⋅exp(y).