This is an old version, view current version.

## 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 lound 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

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