Gaussian Entropy Power and Independence
See why independent Gaussian sums saturate an entropy bound while correlated sums need not.
Free theory articleSee why independent Gaussian sums saturate an entropy bound while correlated sums need not.
Concept & interpretation
The entropy power of a random variable is the variance of a standard Gaussian that has the same differential entropy. For independent Gaussian variables, the entropy power of their sum is equal to the sum of their entropy powers, establishing an important equality condition for the Entropy Power Inequality (EPI). When variables are correlated, this relationship changes significantly.
Experiment
- Set First standard deviation to 1, Second to 2, and Correlation to 0, observing the Entropy power sum matches the Independent entropy-power bound at 5.
- Change Correlation to 1 and observe the Entropy power sum increases to 9.
Notice that negative correlation can violate the numeric sum condition without breaking the independent bound. Open the Interactive Module