Average pairwise overlap of a Gaussian mixture

Returns MixSim's BarOmega: the unweighted mean of pairwise overlaps $$ \overline{\omega} = \frac{2}{J(J-1)} \sum_{1\le j<\ell\le J} \bigl(\Omega_{j\ell}+\Omega_{\ell j}\bigr) \in[0,1] $$ (up to the MixSim numerical convention), where \(\Omega_{j\ell}=\Pr_{X\sim f_j}(X\text{ classified as }\ell)\) already uses the mixture weights \(\boldsymbol{p}\) inside the Bayes / MAP rule of MixSim::overlap(). Do not multiply the directional masses by \(p_j\) again.

Usage

compute_average_overlap(true_theta, J = NULL)

Arguments

true_theta

List validated by check_true_theta(): p (length \(J\) or \(J\times N\)), mu (\(G\times J\)), sigma (\(G\times G\times J\)).

J

Number of cell types (components). Defaults to the third dimension of sigma.

Value

Scalar average pairwise overlap (MixSim BarOmega).

See also

check_true_theta()

Examples

set.seed(1)
theta <- list(
  p = c(0.5, 0.5),
  mu = cbind(c(0, 0), c(3, 0)),
  sigma = array(c(diag(2), diag(2)), dim = c(2, 2, 2))
)
compute_average_overlap(theta)
#> [1] 0.1336144