Marginal Mean of rho

Computes E(rho | a, b) when alpha ~ Gamma(a, b) (shape-rate).

Usage

mean_rho(a, b, M = .QUAD_NODES_DEFAULT)

Arguments

a

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

b

Numeric; rate parameter of the Gamma prior on alpha (b > 0).

M

Integer; number of quadrature nodes. Default is 80.

Value

Numeric; E(rho | a, b).

Details

Uses Gauss-Laguerre quadrature via integrate_gamma. A key identity is E(rho | alpha) = E(w1 | alpha) = 1/(1+alpha), so E(rho | a, b) equals E(w1 | a, b) (but the full distributions differ).

Interpretation:

• E(rho) > 0.5: High prior co-clustering probability

• E(rho) in (0.2, 0.5): Moderate co-clustering

• E(rho) < 0.2: Low co-clustering (fragmented prior)

References

Lee, J. (2025). RN-06: Dual-Anchor Design II, Section 3.3.

See also

var_rho, cv_rho, mean_w1

Examples

mean_rho(a = 2, b = 1)
#> [1] 0.4036526
mean_rho(a = 1.6, b = 1.22)
#> [1] 0.508368