Computes E(w₁ | a, b) via Gauss-Laguerre quadrature.
Usage
mean_w1(a, b, M = .QUAD_NODES_DEFAULT)
Arguments
- a
Numeric; shape parameter of the Gamma prior on α (a > 0).
- b
Numeric; rate parameter of the Gamma prior on α (b > 0).
- M
Integer; number of quadrature nodes. Default is 80.
Details
The expectation is computed using the identity:
$$E[w_1 | a, b] = E\left[\frac{1}{1+\alpha}\right] = I_1(a, b)$$
where the integral is evaluated via Gauss-Laguerre quadrature.
Key identity: E(w₁ | a, b) = E(ρ | a, b) where ρ = Σwₕ² is
the co-clustering probability.
References
Lee, J. (2025). RN-06: Dual-Anchor Design II, §2.3.
Examples
mean_w1(a = 2, b = 1) # ~0.404
#> [1] 0.4036526
mean_w1(a = 1.6, b = 1.22) # ~0.508
#> [1] 0.508368
