Read the equation
For each nontrivial split, the numerator compares the two within-part conditional entropies with the whole-system conditional entropy. K is the smaller marginal entropy, J is their ratio, and S* minimizes that ratio. The benchmark saves both the dimensionless minimized ratio and the raw numerator in nats; this observational measure is not the full IIT 3.0 or 4.0 formalism.
Terms & definitions
| Term | Meaning in this source |
|---|---|
| S | Nonempty proper subset |
| S_bar | Complement |
| Phi_SI | Stochastic-interaction numerator |
| K | Positive normalization denominator |
| J | Optimized ratio |
| S_star | Minimizing partition |
How it works
Choose a nonempty proper subset S and its complementary block.
Compute the stochastic-interaction numerator Φ_SI(S) and the smaller marginal-entropy normalization K(S).
For positive K(S), form J(S)=Φ_SI(S)/K(S).
Search for the partition minimizing J. Report the normalized objective separately from the unnormalized numerator at the selected cut.
Assumptions & limits
Differential entropy and the selected partition depend on variable units; the benchmark fixes innovation scale.
Original context
Read the surrounding source passage
We solve for the stationary population covariance analytically. Thus the experiment tests partition search without finite-sample covariance-estimation error. Root seed: \texttt{20261012}; all matrices and seed derivations are retained.
\section*{3. Exact normalized objective}
For a nonempty proper subset $S$ and complement $\bar S$, define [1,2]
\begin{align*}
\Phi_{\rm SI}(S)&=H(X_{t+1,S}\mid X_{t,S})+
H(X_{t+1,\bar S}\mid X_{t,\bar S})-H(X_{t+1}\mid X_t),\\
K(S)&=\min\{H(X_{t,S}),H(X_{t,\bar S})\},\qquad
J(S)=\Phi_{\rm SI}(S)/K(S),\qquad S^*=\arg\min_S J(S).
\end{align*}
The optimized quantity $J$ is dimensionless; the unnormalized $\Phi_{\rm SI}(S^*)$ is separately saved in nats. All logarithms are natural. Differential-entropy normalization depends on variable units: the innovation scale is fixed, not claimed invariant to rescaling. ``Exact'' means exhaustive global search using float64 arithmetic and the tolerance stated on page 2, not symbolic exact arithmetic.
\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*}Exact original notation
\Phi_{\rm SI}(S)&=H(X_{t+1,S}\mid X_{t,S})+
H(X_{t+1,\bar S}\mid X_{t,\bar S})-H(X_{t+1}\mid X_t),\\
K(S)&=\min\{H(X_{t,S}),H(X_{t,\bar S})\},\qquad
J(S)=\Phi_{\rm SI}(S)/K(S),\qquad S^*=\arg\min_S J(S).Related builds
Sources & provenance
\Phi_{\rm SI}(S)&=H(X_{t+1,S}\mid X_{t,S})+ H(X_{t+1,\bar S}\mid X_{t,\bar S})-H(X_{t+1}\mid X_t),\\ K(S)&=\min\{H(X_{t,S}),H(X_{t,\bar S})\},\qquad J(S)=\Phi_{\rm SI}(S)/K(S),\qquad S^*=\arg\min_S J(S).
\Phi_{\rm SI}(S)&=H(X_{t+1,S}\mid X_{t,S})+ H(X_{t+1,\bar S}\mid X_{t,\bar S})-H(X_{t+1}\mid X_t),\\ K(S)&=\min\{H(X_{t,S}),H(X_{t,\bar S})\},\qquad J(S)=\Phi_{\rm SI}(S)/K(S),\qquad S^*=\arg\min_S J(S).