Mean-Linearization Error Bound

Computes an upper bound on the TV distance between two Poisson distributions, \(\text{Poisson}(\lambda_J(\alpha))\) and \(\text{Poisson}(\alpha c_J)\), using the Poisson-Poisson KL divergence together with Pinsker's inequality.

Usage

compute_linearization_bound(J, alpha, cJ = log(J))

Arguments

J

Integer; sample size.

alpha

Numeric; concentration parameter (vectorized).

cJ

Numeric; scaling constant (default: log(J)).

Value

Numeric; upper bound via Pinsker's inequality.

Details

Let \(\lambda = \lambda_J(\alpha)\) (exact shifted mean) and \(\lambda' = \alpha c_J\) (A1 approximate mean).

The KL divergence is: $$KL(\text{Poisson}(\lambda) || \text{Poisson}(\lambda')) = \lambda \log(\lambda/\lambda') + \lambda' - \lambda$$

By Pinsker's inequality: $$d_{TV}(\text{Poisson}(\lambda), \text{Poisson}(\lambda')) \le \sqrt{KL/2}$$

Numerical safeguards handle edge cases where \(\lambda\) or \(c_J\) is zero.

References

RN-05, Lemma 1: Poisson-Poisson TV bound via KL + Pinsker

See also

compute_poissonization_bound, compute_total_tv_bound

Examples

# Linearization bound for J=50, alpha=1
compute_linearization_bound(J = 50, alpha = 1)
#> [1] 0.1062796

# Effect of J on linearization bound (should decrease)
sapply(c(25, 50, 100, 200), function(J)
  compute_linearization_bound(J, alpha = 2))
#> [1] 0.3579663 0.3328078 0.3098929 0.2899216