February 10, 2016

Logistic Transformation

Sometimes transformations can have unintended results. For example, consider a weakly informative Gaussian density transformed to the unit interval using the standard logistic transformation. \[y=\frac{1}{1+e^{-x}}\] Where \(X\) is a Gaussian random variable. The support of \(Y\) will be on the unit interval as desired, but the shape of the transformed density can have unexpected behavior.

Gaussian Densities with different variance parameters

Transformed to Unit Interval