Average pairwise Jeffreys divergence of a Gaussian mixture

Returns the \(p_j p_{\ell}\)-weighted mean of pairwise Jeffreys (symmetrised KL) divergences between purified Gaussians $$ \overline{J} = \frac{ \sum_{1\le j<\ell\le J} p_j p_{\ell}\, J(f_{j},f_{\ell}) }{ \sum_{1\le j<\ell\le J} p_j p_{\ell} } \in[0,\infty), $$ with \(f_{j}=\mathcal{N}_{G}( \boldsymbol{\mu}_{\cdot j},\boldsymbol{\Sigma}_{j} )\). If p is omitted it defaults to the equi-balanced vector \(1/J\), which recovers the uniform pairwise average \(2/(J(J-1))\sum_{j<\ell}J(f_{j},f_{\ell})\).

Usage

compute_average_jeffreys(true_theta, J = NULL)

Arguments

true_theta

List validated by check_true_theta(): mu (\(G\times J\)), sigma (\(G\times G\times J\)), and optionally p (length \(J\) or \(J\times N\)). If p is missing it is set to \((1/J,\ldots,1/J)\).

J

Number of cell types. Defaults to the third dimension of sigma.

Value

Scalar average pairwise Jeffreys divergence.

See also

check_true_theta(), compute_average_overlap(), compute_glmnet_gene_scores()

Examples

set.seed(1)
theta <- list(
  mu = cbind(c(0, 0), c(3, 0)),
  sigma = array(c(diag(2), diag(2)), dim = c(2, 2, 2))
)
compute_average_jeffreys(theta)
#> [1] 9