Conditional Variance of rho Given Alpha

Computes the conditional variance Var(rho | alpha).

Usage

var_rho_given_alpha(alpha)

Arguments

alpha

Numeric vector; concentration parameter(s) (must be positive).

Value

Numeric vector; Var(rho | alpha).

Details

The conditional variance is: $$Var(\rho | \alpha) = \frac{2\alpha}{(1+\alpha)^2(2+\alpha)(3+\alpha)}$$

This is derived from the GEM recursion: rho = V^2 + (1-V)^2 * rho' where V ~ Beta(1, alpha) and rho' is an independent copy of rho.

Properties:

• Var(rho|alpha) = 0 when alpha -> 0 (degenerate at rho = 1)

• Var(rho|alpha) -> 0 when alpha -> Inf (degenerate at rho = 0)

• Maximum variance occurs at intermediate alpha

References

Lee, J. (2025). RN-06: Dual-Anchor Design II, Appendix A.2.

See also

mean_rho_given_alpha, var_rho

Examples

var_rho_given_alpha(2)
#> [1] 0.02222222
var_rho_given_alpha(c(0.5, 1, 2, 5, 10))
#> [1] 0.050793651 0.041666667 0.022222222 0.004960317 0.001059547