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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

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