Total TV Error Bound (Conditional)

Computes the combined conditional TV bound using the triangle inequality: $$d_{TV}(K_J | \alpha, 1 + \text{NegBin}) \le B_{\text{Pois}} + B_{\text{lin}}$$

Usage

compute_total_tv_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 on total TV error (capped at 1).

Details

The result is capped at 1 since TV distance is bounded by 1.

From RN-05, Theorem 1, the total conditional TV error decomposes as:

1. Poissonization error: \(S_J | \alpha\) vs \(\text{Poisson}(\lambda_J(\alpha))\)

2. Linearization error: \(\text{Poisson}(\lambda_J(\alpha))\) vs \(\text{Poisson}(\alpha c_J)\)

See also

compute_poissonization_bound, compute_linearization_bound, expected_tv_bound

Examples

# Total bound at alpha = E[alpha] under Gamma(2, 1)
compute_total_tv_bound(J = 50, alpha = 2)
#> [1] 0.5809985

# Vectorized
compute_total_tv_bound(J = 50, alpha = c(0.5, 1, 2, 5))
#> [1] 0.1075542 0.2795305 0.5809985 1.0000000