Bennett Bernoulli Tail Budget
Compare rare-event probabilities with a bound that uses both variance and support.
Free theory articleCompare rare-event probabilities with a bound that uses both variance and support.
Concept & interpretation
The Bennett inequality is a concentration bound for sums of independent random variables. Unlike simpler bounds like Hoeffding, which only depend on the bounded range of the variables, Bennett's inequality utilizes the variance budget. For a sum of Bernoulli random variables (a Binomial distribution), this produces a strictly tighter bound when the variance is small compared to the support limit.
Experiment
In the interactive module, you can select the total Trials, the Success probability, and the Tail threshold. The visualizer calculates the exact upper tail, the variance budget, the centered threshold t = k - np, and applies Bennett's upper bound formula. You will see on the chart how the Bennett bound is influenced by the limited variance budget, keeping probabilities significantly lower than bounds that ignore variance. To use the module:
- Change the Trials and Success probability controls to alter the binomial distribution.
- Adjust the Tail threshold to observe how the exact upper tail probability compares to the Bennett bound on the visualization.