Bhatia–Davis Variance Bound

Worked Example

Consider a random variable X constrained to the interval [0, 1]. Suppose we know its expected value is exactly E[X] = μ = 0.25. Let us construct a specific probability distribution with this mean to explore its variance.

We construct a mixture distribution: with probability s, X is drawn from a Bernoulli(μ) distribution concentrated on the extreme endpoints {0, 1}; with probability 1-s, X takes the constant value μ exactly.

If s = 0.5:

1. The mass at 0 is s * (1 - μ) = 0.5 * (1 - 0.25) = 0.375
2. The mass at μ is 1 - s = 1 - 0.5 = 0.5
3. The mass at 1 is s * μ = 0.5 * 0.25 = 0.125

The expected value is E[X] = 0*0.375 + 0.25*0.5 + 1*0.125 = 0.125 + 0.125 = 0.25, which matches our target μ.

To find the variance, we calculate E[X²] - (E[X])²:

We can derive the Bhatia-Davis bound generally for any X in [0, 1] by noting that X(1-X) >= 0 always holds. Taking the expectation gives E[X(1-X)] >= 0, which expands to E[X] - E[X²] >= 0, meaning E[X²] <= μ. Since Var(X) = E[X²] - μ², we immediately get Var(X) <= μ - μ² = μ(1-μ).

E[X²] = 0²*0.375 + 0.25²*0.5 + 1²*0.125 = 0 + 0.0625*0.5 + 0.125 = 0.03125 + 0.125 = 0.15625 Var(X) = E[X²] - 0.25² = 0.15625 - 0.0625 = 0.09375

Alternatively, the formula for the variance of this mixture is s * μ * (1 - μ). Here, 0.5 * 0.25 * 0.75 = 0.09375.

The Bhatia–Davis bound gives a maximum variance of μ * (1 - μ) = 0.25 * 0.75 = 0.1875. The Popoviciu bound is simply 1/4 = 0.25. The actual variance (0.09375) is strictly less than the Bhatia-Davis bound, meaning there is "slack" of 0.1875 - 0.09375 = 0.09375.

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.

Limitations

The Bhatia–Davis inequality assumes strict bounds on the domain of the random variable (e.g., [m, M], scaled to [0, 1] in this module). If the random variable can take values outside this declared bounding interval, the inequality no longer holds. The declared bounding interval does not need to be the smallest actual support, but it must strictly contain all possible values. Furthermore, while the bound is sharp (equality can be achieved by placing all probability mass on the endpoints), many real-world distributions are unimodal and concentrated near their mean, resulting in variances significantly lower than the Bhatia-Davis limit.

References

For more information on the mathematical proof and related inequalities, please see the primary literature:

1. Bhatia, R., & Davis, C. (2000). A Better Bound on the Variance. The American Mathematical Monthly, 107(4), 353-357. https://www.tandfonline.com/doi/abs/10.1080/00029890.2000.12005203