Le Cam Rare-Event Limit
Quantify a rare-event Poisson approximation with an exact overlap distance and finite bound.
Free theory articleQuantify a rare-event Poisson approximation with an exact overlap distance and finite bound.
Concept & interpretation
When modeling the number of successes in a large number of independent trials, each with a small probability of success, the Binomial distribution can be well approximated by a Poisson distribution with the same mean. Le Cam's inequality provides a concrete bound on how large the discrepancy between these two distributions can be.
Testing the convergence
The Le Cam Rare-Event Limit explorer plots the Binomial and Poisson distributions, enabling direct visual and numerical comparison.
- Set Fixed Poisson mean to 1, and Trials to 4. Check the Total Variation which matches our 0.08099 calculation.
- Increase Trials to 16. The Bernoulli probability drops to 0.0625, the Le Cam bound drops to 0.0625, and the true Total Variation distance tightens to 0.01789.
- The visualization clearly shows the Binomial bars converging closely around the Poisson curve points.
- Adjust the Count probe to compare exact pointwise probabilities between the models.