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Variance of w₁

Source: R/08_weights_w1.R
var_w1.Rd

Computes Var(w₁ | a, b) using the law of total variance.

Usage

var_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.

Value

Numeric; Var(w₁).

Details

Uses the law of total variance: $$Var(w_1) = E[Var(w_1 | \alpha)] + Var(E[w_1 | \alpha])$$

where w₁ | α ~ Beta(1, α), so:

  • E(w₁ | α) = 1/(1+α)

  • Var(w₁ | α) = α / ((1+α)²(2+α))

References

Lee, J. (2026). Design-Conditional Prior Elicitation for Dirichlet Process Mixtures. arXiv preprint arXiv:2602.06301.

See also

mean_w1, summary_w1

Other weights_w1: cdf_w1(), density_w1(), mean_w1(), prob_w1_exceeds(), quantile_w1(), summary_w1()

Examples

var_w1(a = 2, b = 1)       # ~0.090
#> [1] 0.08968429
var_w1(a = 1.6, b = 1.22)  # ~0.105
#> [1] 0.1052062