Survival Function of w₁

Computes P(w₁ > t | a, b) = 1 - F(t), the probability that the first stick-breaking weight exceeds threshold t.

Usage

prob_w1_exceeds(t, a, b)

Arguments

t: Numeric vector of thresholds. Values outside the unit interval are allowed but are mapped to the boundary values (1 for t ≤ 0, 0 for t ≥ 1).
a: Numeric; shape parameter of the Gamma prior on α (a > 0).
b: Numeric; rate parameter of the Gamma prior on α (b > 0).

Value

Numeric vector of survival probabilities.

Details

The survival function has the closed form: $$P(w_1 > t | a, b) = \left(\frac{b}{b - \log(1-t)}\right)^a$$

This is a key quantity for dominance risk assessment (RN-07). A large P(w₁ > 0.5) indicates high prior probability that a single cluster dominates the mixture.

Dominance Risk Interpretation

P(w₁ > 0.5) ≈ 0.5: moderate dominance risk
P(w₁ > 0.9) ≈ 0.1: low extreme dominance risk

References

Lee, J. (2025). RN-06: Dual-Anchor Design II, Corollary 1. Lee, J. (2025). RN-07: Unintended Prior Consequences.

Examples

# P(w₁ > 0.5): "dominant cluster" probability
prob_w1_exceeds(0.5, a = 1.6, b = 1.22)  # ~0.487 (Lee et al. DP-inform)
#> [1] 0.4868311
prob_w1_exceeds(0.5, a = 2, b = 1)       # ~0.349
#> [1] 0.3488274