Gaussian relative entropy

Relative entropy of one multivariate Gaussian from another, optionally split into its signal and dispersion parts.

Usage

aci_metric_pair(mu_p, R_p, mu_q, R_q, decompose = TRUE)

Arguments

mu_p

Numeric vector, mean of the first distribution.

R_p

Covariance matrix of the first distribution.

mu_q

Numeric vector, mean of the second distribution.

R_q

Covariance matrix of the second distribution.

decompose

TRUE to return the signal and dispersion parts alongside the total.

Value

A named numeric vector with the total and, when decompose is TRUE, the signal and dispersion parts.

Details

Public KL values never apply a covariance ridge; callers wanting regularisation opt in explicitly with spd_floor(). andreou2026cir (Section 2.2, closing paragraph) states that regularization is expected only in the degenerate limit, which is the published basis for this function's strictness. The state recursions are strict by default too, and regularise only when a call asks for it with regularize = "floor"; see aci_filter().

The dispersion part is O(delta^2) in delta = R_p / R_q - 1, and this function evaluates it in the trace and log-difference form 0.5 * (tr(R_q^-1 R_p) - l + log det R_q - log det R_p) at every dimension, including l = 1. That form loses relative precision to cancellation as the two covariances approach each other: measured against the exact series at R_q = 1, R_p = 1 + delta, its relative error is 4.5e-05 at delta = 1e-06, 1.2e-02 at 1e-07 and a factor of two from 1e-08 down. aci_metric() instead routes a one-dimensional hidden state through a cancellation-resistant log1p form, so the two public routes disagree on identical one-dimensional inputs in that regime. The absolute error stays below 3e-17 nats across that sweep, because the quantity itself is quadratic in delta, so this is a precision boundary and not a demonstrated accuracy failure on ordinary inputs. Positivity is imposed by max(., 0) on each part; that is a guard, not an accuracy proof.

See also

aci_metric(), aci()

Examples

aci_metric_pair(mu_p = 0, R_p = matrix(1), mu_q = 1, R_q = matrix(2))
#>      total     signal dispersion 
#> 0.34657359 0.25000000 0.09657359