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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
\[ \left| \, \frac{d}{dy} \left( \exp(y) + a \right) \, \right| = \exp(y). \]
Therefore, given the density \(p_X\) of \(X\), the density of \(Y\) is
\[ p_Y(y) = p_X\!\left( \exp(y) + a \right) \cdot \exp(y). \]