10.2 Lower Bounded Scalar

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Stan uses a logarithmic transform for lower and upper bounds.

If a variable $X$ is declared to have lower bound $a$ , it is transformed to an unbounded variable $Y$ , where

$Y = \log(X - a).$

The inverse of the lower-bound transform maps an unbounded variable $Y$ to a variable $X$ that is bounded below by $a$ by

$X = \exp(Y) + a.$

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

$p_Y(y) = p_X\!\left( \exp(y) + a \right) \cdot \exp(y).$