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Fix #356 - affine transform docs has the wrong direction of transform #357

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May 12, 2021
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Fix #356 - affine transform docs has the wrong direction of transform #357

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12 changes: 6 additions & 6 deletions src/reference-manual/transforms.Rmd
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Original file line number Diff line number Diff line change
Expand Up @@ -292,7 +292,7 @@ Stan uses an affine transform. Such a variable $X$ is transformed to a new
variable $Y$, where

$$
Y = \mu + \sigma * X.
Y = \frac{X - \mu}{\sigma}.
$$

The default value for the offset $\mu$ is $0$ and for the multiplier $\sigma$ is
Expand All @@ -304,7 +304,7 @@ $1$ in case not both are specified.
The inverse of this transform is

$$
X = \frac{Y-\mu}{\sigma}.
X = \mu + \sigma \cdot Y.
$$

### Absolute derivative of the affine inverse transform
Expand All @@ -316,19 +316,19 @@ $$
\left|
\frac{d}{dy}
\left(
\frac{y-\mu}{\sigma}
\mu + \sigma \cdot y
\right)
\right|
= \frac{1}{\sigma}.
= \sigma.
$$

Therefore, the density of the transformed variable $Y$ is

$$
p_Y(y)
=
p_X \! \left( \frac{y-\mu}{\sigma} \right)
\cdot \frac{1}{\sigma}.
p_X \! \left( \mu + \sigma \cdot y \right)
\cdot \sigma.
$$


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