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Average pairwise overlap of a Gaussian mixture

Source: R/statistical_metrics.R
compute_average_overlap.Rd

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.

For \(G\ge 4\) MixSim's Davies quadrature becomes expensive. The helper then switches to overlap_gaussian_mc(): stratified Sobol draws, inverse-transform sampling through precomputed Cholesky factors, and log-density MAP comparisons (n_mc draws per component; default 10,000).

Usage

compute_average_overlap(
  true_theta,
  J = NULL,
  n_mc = 10000L,
  seed = NULL,
  verbose = TRUE
)

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.

n_mc

Monte Carlo draws per component when \(G\ge 4\).

seed

Optional seed forwarded to overlap_gaussian_mc().

verbose

If TRUE (default), announce the MixSim-to-MC switch with cli when it is installed.

Value

Scalar average pairwise overlap (MixSim BarOmega).

See also

check_true_theta(), overlap_gaussian_mc()

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