Bhatia–Davis Variance Bound

The Bhatia–Davis inequality provides a sharp upper bound on the variance of any bounded random variable, given its mean. Unlike the general Popoviciu's inequality on variances, which states that Var(X) <= 1/4 for any X in [0, 1], the Bhatia–Davis bound incorporates knowledge of the expected value to yield a strictly tighter or equal bound.

Experiment

You can interactively explore the Bhatia-Davis variance bound by adjusting the parameters of the mixture distribution in the interactive module.

1. Mean μ: Controls the fixed expected value of the distribution (0, 0.25, 0.5, 0.75, or 1). Observe how changing the mean affects the Bhatia-Davis bound curve μ(1-μ). When μ approaches 0 or 1, the maximum possible variance approaches 0.
2. Endpoint share (s): Controls how much probability mass is pushed to the extreme endpoints {0, 1} versus the center μ. Increasing the endpoint share increases the variance. When s = 1, all mass is at the boundaries, the variance hits the Bhatia-Davis bound, and equality holds.

Notice how the Bhatia-Davis bound is always tighter than or equal to the Popoviciu bound (0.25), providing a more constrained maximum variance when the mean is known to be away from the center of the interval.