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Conditional Gaussian entropy is determined by a conditional covariance log determinant. With full-system innovation covariance equal to identity, its log determinant vanishes and the stochastic-interaction numerator reduces to half the sum of the two partition residual log determinants. Marginal entropy includes both dimension and covariance scale, so the resulting normalization is sensitive to variable units.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| R_S | Residual covariance for predicting subset S from its own previous state |
| Sigma_SS | Stationary covariance of subset S |
| S | Nonempty proper subset |
| Phi_SI | Stochastic-interaction loss in nats |
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\newpage
\section*{4. Computation and comparison protocol}
\subsection*{Gaussian evaluation and exhaustive reference}
Let $G=W\Sigma$ and $R_S=\Sigma_{SS}-G_{SS}\Sigma_{SS}^{-1}G_{SS}^{\top}$. The implementation uses Cholesky solves and log-determinants, not explicit matrix inversion:
\begin{equation*}
\Phi_{\rm SI}(S)=\tfrac12[\log|R_S|+\log|R_{\bar S}|],\qquad
H(X_{t,S})=\tfrac12[|S|\log(2\pi e)+\log|\Sigma_{SS}|].
\end{equation*}
The full-system innovation determinant is one. We pin node 1 to $S$ to remove complement duplicates, leaving $2^{n-1}-1$ valid bipartitions: 127, 511, 2,047 and 8,191 per instance. Enumeration records every $J$, numerator, denominator and optimal partition. Recovery requires
$J_{\rm returned}\le J^*+10^{-10}+10^{-8}|J^*|$. No tied optima occurred at this tolerance.
\subsection*{Queyranne: direct normalized-objective search}
We use pendant-pair contraction with a deterministic starting order and cache objective values within each search. The minimum candidate is returned without an added local-search step. Queyranne has a guarantee for symmetric submodular functions [3]; that guarantee is not assumed for normalized stochastic interaction. Consequently, its outputs are judged against enumeration, not treated as reference answers. Each run has a fresh on-demand oracle and cannot read the exhaustive table.
\subsection*{Simulated annealing: independently repeated searches}Exact original notation
\Phi_{\rm SI}(S)=\tfrac12[\log|R_S|+\log|R_{\bar S}|],\qquad
H(X_{t,S})=\tfrac12[|S|\log(2\pi e)+\log|\Sigma_{SS}|].Related builds
Sources & provenance
\Phi_{\rm SI}(S)=\tfrac12[\log|R_S|+\log|R_{\bar S}|],\qquad H(X_{t,S})=\tfrac12[|S|\log(2\pi e)+\log|\Sigma_{SS}|].
\Phi_{\rm SI}(S)=\tfrac12[\log|R_S|+\log|R_{\bar S}|],\qquad H(X_{t,S})=\tfrac12[|S|\log(2\pi e)+\log|\Sigma_{SS}|].