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→冗余信息计算方法
比如
比如
\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 $S$.
For a collection of sources $\alpha \in \mathcal{A}(\mathbf{R})$, the PI-function, denoted $\Pi_{\mathbf{R}}$, is defined implicitly by
\begin{equation}
I_{\min}(S; \alpha) = \sum_{\beta \prec \alpha} \Pi_{\mathbf{R}}(S; \beta).
\end{equation}
Formally, $\Pi_{\mathbf{R}}$ corresponds to the Möbius inverse of $I_{\min}$ \cite{rota1964foundations, stanley1997enumerative}. From this relationship, it is clear that $\Pi_{\mathbf{R}}$ can be calculated recursively as
\begin{equation}
\Pi_{\mathbf{R}}(S; \alpha) = I_{\min}(S; \alpha) - \sum_{\beta \prec \alpha} \Pi_{\mathbf{R}}(S; \beta).
\end{equation}
Put into words, $\Pi_{\mathbf{R}}(S; \alpha)$ quantifies the information provided redundantly by the sources of $\alpha$ that is not provided by any simpler collection of sources (i.e., any $\beta$ lower than $\alpha$ on the redundancy lattice). In Appendix D, it is shown that $\Pi_{\mathbf{R}}$ can be written in closed form as
\begin{equation}
\Pi_{\mathbf{R}}(S; \alpha) = I_{\min}(S; \alpha) - \sum_{s} p(s) \max_{\beta \in \alpha^{-}} \min_{\mathbf{B} \in \beta} I(S = s; \mathbf{B})
\end{equation}
where $\alpha^{-}$ represents the nodes immediately below $\alpha$ in the redundancy lattice. From this formulation, it is readily shown that $\Pi_{\mathbf{R}}$ is nonnegative (see Appendix D), and thus can be naturally
== 延伸与应用 ==
== 延伸与应用 ==