Tail Probability of the First Size-Biased DP Weight

Computes P(W_SB > t | a, b) = 1 - F(t), the probability that a randomly selected unit belongs to a population cluster whose DP mass exceeds 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 \(t \le 0\), 0 for \(t \ge 1\)).

a

Numeric; shape parameter of the Gamma prior on \(\alpha\) (a > 0).

b

Numeric; rate parameter of the Gamma prior on \(\alpha\) (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 mass-weighted cluster-size probability. It is not \(P(W_{max}>t)\), the probability that the largest population weight exceeds \(t\). Use prob_wmax_exceeds for that estimand.

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(), prob_wmax_exceeds(), prob_wsb_exceeds(), quantile_w1(), summary_w1(), var_w1(), wmax_tail_bounds()

Examples

# Probability that a random unit's population cluster has mass above 0.5
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