Read the equation
The expression subtracts the variance explained by the subset’s own previous state from its next-state marginal covariance. It is a Schur complement and is the conditional covariance of Xₜ₊₁,S given Xₜ,S. The implementation evaluates it through linear solves rather than constructing the matrix inverse explicitly.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| R_S | Subset residual covariance |
| Sigma_SS | Subset stationary covariance |
| G_SS | Subset cross-covariance |
How it works
Read the definition from its right-hand expression to the quantity on the left. Its surrounding source records the intended input, state, or parameter context. Where the source leaves an operation unspecified, this page preserves that gap.
Assumptions & limits
Source recovery establishes that the expression appears in this record. Its mathematical validity, implementation behavior, and interpretation must be assessed separately. See the linked source and research notes for the checks available for this version.
Original context
Read the surrounding source passage
\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.Exact original notation
R_S=\Sigma_{SS}-G_{SS}\Sigma_{SS}^{-1}G_{SS}^{\top}