Conditional Mean of rho Given Alpha

Computes the conditional mean E(rho | alpha) for the co-clustering probability rho = sum_h w_h^2 under a Dirichlet Process.

Usage

mean_rho_given_alpha(alpha)

Arguments

alpha

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

Value

Numeric vector; E(rho | alpha) = 1/(1+alpha).

Details

The co-clustering probability rho = sum(w_h^2) over h >= 1 has conditional mean: $$E[\rho | \alpha] = \frac{1}{1 + \alpha}$$

This equals E(w1 | alpha) since w1 ~ Beta(1, alpha) has mean 1/(1+alpha).

Interpretation:

• alpha -> 0: E(rho|alpha) -> 1 (all observations in one cluster)

• alpha -> Inf: E(rho|alpha) -> 0 (infinitely many small clusters)

• alpha = 1: E(rho|alpha) = 0.5 (moderate clustering)

References

Lee, J. (2025). RN-06: Dual-Anchor Design II, Section 3.2.

See also

var_rho_given_alpha, mean_rho

Examples

mean_rho_given_alpha(1.0)
#> [1] 0.5
mean_rho_given_alpha(c(0.5, 1, 2, 5, 10))
#> [1] 0.66666667 0.50000000 0.33333333 0.16666667 0.09090909