The incorrect formula was used in the document and comments.
|
Logit-normal |
Log-normal |
| Normal variable |
$X \sim \mathcal{N}(\mu, \sigma^2)$ |
$X \sim \mathcal{N}(\mu, \sigma^2)$ |
| Transformation |
$Y = \dfrac{1}{1+e^{-X}}$ |
$Y = e^X$ |
| Inverse transformation |
$X = \ln\dfrac{Y}{1-Y}$ |
$X = \ln Y$ |
| Support |
$(0, 1)$ |
$(0, +\infty)$ |
| Jacobian |
$\dfrac{1}{y(1-y)}$ |
$\dfrac{1}{y}$ |
| PDF |
$\dfrac{1}{y(1-y)} \cdot \dfrac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\dfrac{\left(\ln\frac{y}{1-y} - \mu\right)^2}{2\sigma^2}\right)$ |
$\dfrac{1}{y} \cdot \dfrac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\dfrac{(\ln y - \mu)^2}{2\sigma^2}\right)$ |
The correct Logit-normal pdf should be:
$$
f(x; \mu, \sigma) = \dfrac{1}{x(1-x)} \cdot \dfrac{1}{\sqrt{2\pi\sigma^2}} \exp\left(-\dfrac{\left(\text{logit}(x) - \mu\right)^2}{2\sigma^2}\right)
$$
Instead of this (written in the comments):
$$
f(x; \mu, \sigma) = \dfrac{1}{x\sqrt{2\pi\sigma^2}} \exp\left(-\dfrac{\left(\text{logit}(x) - \mu\right)^2}{2\sigma^2}\right)
$$
The incorrect formula was used in the document and comments.
The correct Logit-normal pdf should be:
Instead of this (written in the comments):