MixSim-style pairwise overlap via stratified Sobol Monte Carlo

Estimates the MixSim Omega map \(\Omega_{j\ell}=\Pr_{X\sim f_j}(\pi_\ell f_\ell(X)>\pi_j f_j(X))\) by drawing n_mc quasi-Monte Carlo points from each component (inverse-transform sampling of a Sobol sequence through the Cholesky factor) and comparing log-densities. The average pairwise overlap BarOmega is the unweighted mean of \(\Omega_{j\ell}+\Omega_{\ell j}\) over \(j<\ell\), matching MixSim::overlap() / FSDA MixSim.

Usage

overlap_gaussian_mc(true_theta, n_mc = 10000L, seed = NULL, J = NULL)

Arguments

true_theta: List with p, mu, sigma as in check_true_theta().
n_mc: Draws per component (default 10000).
seed: Optional RNG seed (passed to Sobol randomisation and to the uniform fallback).
J: Optional number of components.

Value

A list with BarOmega, MaxOmega, and OmegaMap.

Details

For two equal-weight components this pairwise overlap equals the histogram similarity \(\int\min(f_j,f_\ell)\) and therefore \(1-\mathrm{TV}(f_j,f_\ell)\) (Nielsen and Sun 2018) .

References

Nielsen F, Sun K (2018). “Guaranteed Deterministic Bounds on the Total Variation Distance between Univariate Mixtures.” arXiv:1806.11311. doi:10.48550/arxiv.1806.11311 . https://arxiv.org/abs/1806.11311.

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))
)
overlap_gaussian_mc(theta, n_mc = 2000L)$BarOmega
#> [1] 0.1335