Constrained Gaussian-MRF (graph-restricted MLE) stationarity residual

Certificate for an unregularized Gaussian graphical model whose precision is constrained to a fixed graph (the estimator behind ggm_modselect() and logo_network()). The maximum-likelihood / maximum-entropy conditions for a Gaussian Markov random field on a graph \(G\) are exact: \(W_{ij} = S_{ij}\) for every \((i,j)\) on the graph and on the diagonal (\(W = \Theta^{-1}\)), and \(\Theta_{ij} = 0\) for every \((i,j)\) not on the graph. A near-zero return certifies the constrained optimum with no reference solver.

Usage

ggm_support_kkt(theta, cor_matrix, support, active_tol = 1e-08)

Arguments

theta

Precision matrix to test.

cor_matrix

Correlation / covariance the model was fit to.

support

Logical p x p matrix; TRUE where an edge is allowed.

active_tol

Magnitude above which an off-support entry counts as a nonzero violation.

Value

Maximum absolute stationarity violation (scalar); 0 = exact optimum.

Examples

S <- 0.4^abs(outer(1:6, 1:6, "-"))
fit <- ggm_modselect(cor_matrix = S, n = 250)
ggm_support_kkt(fit$precision, S, fit$support)
#> [1] 2.220446e-16