Table 1.

Decompositions of synergistic, unique, and redundant information terms into stochastic and deterministic contributions obtained assuming the weak stochasticity axiom. For each term we show the decompositions resulting from two alternative mutual information partitioning orders (Equation (15)), which are consistent with each other (see Appendix B). For the partitioning order leading to an additive separation of each partial information decomposition (PID) term into a stochastic and deterministic component we also individuate the deterministic contributions ${\Delta }_d(X;\beta )$. Synergy has only a stochastic component, according to the axiom (Equation (13)). Expressions of unique information come from Equations (18) and (A3), and the ones of redundancy from Equations (19) and (A5). The expressions have been simplified with respect to the equations, indicating their form for the case $X\cap i\ne \varnothing$. The terms ${\Delta }_d(X;\beta )$ have analogous expressions for $X\cap j\ne \varnothing$ when a symmetry exists between i and j and are zero otherwise.

Term	Decomposition
$I(X;ij\backslash i,j)$	$I(X-ij;ij\backslash i,j)$
$I(X;i\backslash j)$	$\begin{array}{c}I(X-ij;i\backslash j)+H\left(i\right|j,X-ij)\\ H\left(i\right|j)-I(X-ij;ij\backslash i,j)\end{array}$
$I(X;i.j)$	$\begin{array}{c}I(X-ij;i.j)+I(i;j|X-ij)\\ I(i;j)+I(X-ij;ij\backslash i,j)\end{array}$
Term	Measure
${\Delta }_d(X;ij)$	0
${\Delta }_d(X;i)$	$H\left(i\right|j,X-ij)$
${\Delta }_d(X;i.j)$	$I(i;j|X-ij)$