Conditional-independence (Gaussian graphical model) structure of a mixture

Returns the undirected second-order conditional-independence graph of a fitted Gaussian mixture: the partial-correlation (Gaussian graphical model) structure of the mixture's overall covariance. An edge $i - j$ is present when the partial correlation $r_{ij}$ of coordinates $i$ and $j$ given all the others is judged nonzero, and absent otherwise – the latter is the Markov statement $x_i \perp x_j \mid x_{\mathrm{rest}}$ at second order. The overall covariance $$\mathrm{Cov}(X) = \sum_k w_k (\Sigma_k + \mu_k \mu_k^\top) - \bar\mu\,\bar\mu^\top, \qquad \bar\mu = \sum_k w_k \mu_k,$$ is closed-form in the mixture parameters, so no sampling is required.

Usage

gmm_independence_graph(g, threshold = NULL, alpha = 0.05, n = NULL)

Arguments

g: A gmm (or gmm_fit) mixture.
threshold: Non-negative partial-correlation magnitude above which an edge is drawn. NULL (the default) uses the Fisher-z test when a sample size is available and 0.05 otherwise. Cannot be combined with n.
alpha: Per-edge level of the Fisher-z test, in (0, 1). Used only when a sample size is available and threshold is not given.
n: Number of data points the mixture was fitted to. NULL (the default) takes it from a regime "sample" or "moment" fit and otherwise leaves it unknown.

Value

A symmetric integer adjacency matrix (1 = edge, 0 = none) with the coordinate names of g, carrying the partial-correlation matrix as the "pcor" attribute.

Details

When the mixture was fitted to $n$ data points, each partial correlation is tested with Fisher's z: an edge is drawn when $$|\operatorname{atanh}(r_{ij})| \sqrt{n - p - 1} > \Phi^{-1}(1 - \alpha / 2).$$ A gmm_fit from regime "sample" or "moment" supplies $n$ as the number of rows of its target's samples; n can also be given directly. The test is per edge, and a graph on $p$ coordinates carries $p(p-1)/2$ of them: alpha = 0.05 / choose(p, 2) keeps the chance of any false edge below 0.05. Without a sample size (a regime "kld" fit, or a plain gmm), an edge is drawn when $|r_{ij}|$ exceeds threshold, which defaults to 0.05 there. Giving threshold always selects the magnitude rule.

This is a graphical-model (dependency-structure) diagnostic, not a causal discovery method: it recovers the undirected Markov skeleton, not edge directions. Its distinctive use is regime (iii): composed with fit_kld_em(), it recovers the dependency structure of a target you can only evaluate (an unnormalised energy / Gibbs density), where no sample exists to hand a sampling-based estimator. Being second order, it sees dependencies that enter the covariance; a purely higher-order coupling (zero correlation, nonzero dependence) is not detected – raise the mixture's component count, or read the coordinate-block dependence with gmm_mutual_information() instead.

Examples

## Regime (iii): the Markov structure of an evaluable-but-unsampleable density.
## A continuous chain field x1 - x2 - x3 (couplings only between neighbours).
energy <- function(X) {
  X <- matrix(X, ncol = 3)
  rowSums((X^2 - 1)^2) - 0.7 * (X[, 1] * X[, 2] + X[, 2] * X[, 3])
}
target <- gmm_target(n_dim = 3L, log_density = function(X) -energy(X))
g <- fit_kld_em(target, N = 8L, proposal = is_uniform(3L, -3, 3),
                is_size = 8000L, anneal = TRUE, seed = 1L, support_warn = FALSE)
gmm_independence_graph(g)            # recovers x1 - x2 - x3 (no x1 - x3 edge)
#>    x1 x2 x3
#> x1  0  1  0
#> x2  1  0  1
#> x3  0  1  0
#> attr(,"pcor")
#>            x1        x2         x3
#> x1 1.00000000 0.4793587 0.01689517
#> x2 0.47935872 1.0000000 0.49064194
#> x3 0.01689517 0.4906419 1.00000000