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\section{Partial Information Decomposition}

\section{Partial Information Decomposition}

The redundant information associated with each node of the redundancy lattice includes, but is not limited to, the redundant information provided by all nodes lower in the lattice. Thus, moving from node to node up the lattice, $I_{\min}$ can be thought of as a kind of "cumulative information function," effectively integrating the information provided by increasingly inclusive collections of sources. Next, we derive an inverse of $I_{\min}$ called the partial information function (PI-function). Whereas $I_{\min}$ quantifies cumulative information, the PI-function measures the partial information contributed uniquely by each particular collection of sources. This partial information will form the atoms into which we decompose the total information that $\mathbf{R}$ provides about $S$.

The redundant information associated with each node of the redundancy lattice includes, but is not limited to, the redundant information provided by all nodes lower in the lattice. Thus, moving from node to node up the lattice, I_{\min} can be thought of as a kind of "cumulative information function," effectively integrating the information provided by increasingly inclusive collections of sources. Next, we derive an inverse of I_{\min} called the partial information function (PI-function). Whereas I_{\min} quantifies cumulative information, the PI-function measures the partial information contributed uniquely by each particular collection of sources. This partial information will form the atoms into which we decompose the total information that \mathbf{R} provides about S.

For a collection of sources $\alpha \in \mathcal{A}(\mathbf{R})$, the PI-function, denoted $\Pi_{\mathbf{R}}$, is defined implicitly by

For a collection of sources \alpha \in \mathcal{A}(\mathbf{R}), the PI-function, denoted \Pi_{\mathbf{R}}, is defined implicitly by

\begin{equation}

\begin{equation}

I_{\min}(S; \alpha) = \sum_{\beta \prec \alpha} \Pi_{\mathbf{R}}(S; \beta).

I_{\min}(S; \alpha) = \sum_{\beta \prec \alpha} \Pi_{\mathbf{R}}(S; \beta).