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Survival Function of w₁

Source: R/08_weights_w1.R
prob_w1_exceeds.Rd

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 (Lee, 2026, Section 4). 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. (2026). Design-Conditional Prior Elicitation for Dirichlet Process Mixtures. arXiv preprint arXiv:2602.06301.

See also

cdf_w1, summary_w1

Other weights_w1: cdf_w1(), density_w1(), mean_w1(), quantile_w1(), summary_w1(), var_w1()

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