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10.3 Upper Bounded Scalar

Stan uses a negated logarithmic transform for upper bounds.

Upper Bound Transform

If a variable $X$ is declared to have an upper bound $b$, it is transformed to the unbounded variable $Y$ by

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

Upper Bound Inverse Transform

The inverse of the upper bound transform converts the unbounded variable $Y$ to the variable $X$ bounded above by $b$ through

$$X = b - \exp(Y).$$

Absolute Derivative of the Upper Bound Inverse Transform

The absolute derivative of the inverse of the upper bound transform is

$$\left| \, \frac{d}{dy} \left( b - \exp(y) \right) \, \right| = \exp(y).$$

Therefore, the density of the unconstrained variable $Y$ is defined in terms of the density of the variable $X$ with an upper bound of $b$ by

$$p_Y(y) = p_X \!\left( b - \exp(y) \right) \cdot \exp(y).$$