GaussianCoupling node

The GaussianCoupling node encodes the stochastic pairwise Gaussian coupling (a bilinear interaction)

\[\phi(\mathrm{out}, \mathrm{in}, a) = \exp(\mathrm{out} \cdot a \cdot \mathrm{in})\]

This is the pairwise (edge) potential of Gaussian Belief Propagation (GaBP) as formulated by Shental et al., Gaussian Belief Propagation for Solving Systems of Linear Equations (arXiv:0810.1119). Their edge potential $\psi_{ij}(x_i, x_j) = \exp(-x_i A_{ij} x_j)$ corresponds to $a = -A_{ij}$, and their self-potential $\phi_i(x_i) = \exp(b_i x_i - A_{ii} x_i^2 / 2)$ is NormalWeightedMeanPrecision(b_i, A[i, i]). Together they turn the solution of $A x = b$ into message passing on a graph.

Interfaces

Factorization

Only the structured factorization q(out, in) q(a) is supported:

@constraints begin
    q(out, in, a) = q(out, in)q(a)
end

When a is supplied as a constant — the intended usage, as in x[j] ~ GaussianCoupling(x[i], -A[i, j]) — this factorization is applied automatically, because constant interfaces are always factorized out of the local cluster. MeanField() is not supported: there are no q(out)q(in) message rules and no mean-field average energy.

Improperness

The potential is not integrable on its own, so it is not a conditional distribution. Note that it is exactly the cross-term of a Gaussian density, since $\mathcal{N}(\mathrm{out}; \mathrm{in}, w^{-1}) \propto \exp(-w \cdot \mathrm{out}^2/2) \exp(w \cdot \mathrm{out} \cdot \mathrm{in}) \exp(-w \cdot \mathrm{in}^2/2)$, with the two quadratic self-terms removed. Consequences:

• The messages towards out and in are improper Gaussians with negative precision. This is inherent to the factor and reproduces the GaBP message $P_{ij} = -A_{ij}^2 / P_{i \setminus j}$.
• The joint marginal q(out, in) is proper only when the incoming precisions dominate the coupling, $w_{\mathrm{out}} w_{\mathrm{in}} > a^2$. This is a local condition on a single factor; it is not the condition under which the method as a whole works — see Accuracy for that.

The Bethe free energy is still meaningful whenever the product of all factors in the model is normalizable. Since $\langle -\log \phi \rangle = -\mathbb{E}[a]\, \mathbb{E}[\mathrm{out} \cdot \mathrm{in}]$ carries no normalizer term, the accumulated score equals $-\log Z + \mathrm{KL}(q \Vert p)$.

Accuracy

Per Shental et al.:

• If A is strictly diagonally dominant, or the diagonally normalized system satisfies $\rho(|I - A|) < 1$, GaBP converges and the inferred means are exact, that is, they solve $A x = b$.
• The inferred variances are exact only on acyclic graphs. On graphs with cycles they are walk-sum approximations of $\mathrm{diag}(A^{-1})$ and must not be reported as the marginal variances of the exact solution.

GaussianCoupling

@constraints begin
    q(out, in, a) = q(out, in)q(a)
end