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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

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