Randomized Probability Integral Transform

Fill the jumps of a discrete CDF with an auxiliary uniform variable to obtain a uniform transform.

Randomized Probability Integral Transform

Concept Overview

Fill the jumps of a discrete CDF with an auxiliary uniform variable to obtain a uniform transform.

Worked Example

Let us walk through an analytical evaluation of the transformation:

Suppose the ordered weights are 1, 2, and 1, resulting in probabilities 1/4, 1/2, and 1/4.
For an observed atom 1, the interval spans from 1/4 to 3/4.
Given a uniform realization of 2/4 (0.5), the randomized transform maps the value to the midpoint 1/2.

Experiment

Try the following steps to observe the uniformization:

Set Weight 0 to 1, Weight 1 to 2, and Weight 2 to 1.
Set Observed atom to 1.
Set Auxiliary quarters to 2.
Set CDF probe quarters to 2.
Observe the transformation filling the discrete jumps.

Explore more at https://riano.app/module/randomized-probability-integral-transform.

Limitations

A true uniform random variable provides infinite precision for the auxiliary variable. We use a discretized fractional auxiliary variable for didactic purposes, which only evaluates the CDF at specific rational coordinates.

References

Primary literature informing this implementation:

Shorack, Probability for Statisticians, Proposition 3.2 - Printed page 109 / PDF page 125 discusses the randomized probability integral transform.

Randomized Probability Integral Transform