Efron–Stein Variance Bound
Bound a nonlinear statistic’s variance by replacing one independent coordinate at a time.
Free theory articleBound a nonlinear statistic’s variance by replacing one independent coordinate at a time.
Concept & interpretation
The Efron–Stein inequality bounds the variance of a general function of independent random variables. It does so by considering how much the function's value can change when a single coordinate is resampled independently.
Experiment
Follow these steps to observe the variance bound in action using the interactive module controls.
- Open the Efron–Stein Variance Bound module.
- Set Function to Sum. Observe that the exact variance and the bound are equal. The Efron-Stein bound is perfectly tight for linear functions like the sum.
- Set Function to AND and Success probability to 0.25. Observe how the bound provides a safe upper estimate for the variance of the minimum function.
- Observe the coordinate resampling visualization. It demonstrates one base sample and compares it to samples where exactly one coordinate is independently redrawn.