更改

James 等（2018）<ref name="James_depcon">{{cite journal|author1=James R. G.|author2=Emenheiser J.|author3=Crutchfield J. P.|title=Unique information via dependency constraints|journal=Journal of Physics A: Mathematical and Theoretical|year=2018|volume=52|issue=1|page=014002|url=https://doi.org/10.1088/1751-8121/aae492}}</ref>将定义冗余信息改为通过依赖约束（dependency constraints）定义特有信息<math>I_{\cap}^{\text{dep}}</math>，要求特有信息分量在边际独立性约束下最大化，给出与<math>I_{\cap}^{∧}</math>不同的另一条路线。Finn等（2018）<ref name="Finn_spec_ambig">{{cite journal|author1=Finn C.|author2=Lizier J. T.|title=Pointwise partial information decomposition using the specificity and ambiguity lattices|journal=Entropy|year=2018|volume=20|issue=4|page=297|url=https://doi.org/10.3390/e20040297}}</ref>将点对点互信息拆分成了正的信息分量（特异性）负的信息分量（模糊性）。与此对应，冗余晶格拆分为了特异性格和模糊性格并据此定义了点对点冗余特异性和点对点冗余模糊性。随后，有两种方式组合这两个定义，一种是直接求期望，一种是从点对点特异性和模糊性中减去点对点特有性、冗余性和协同性，得到点对点特有、冗余和协同信息。Ince（2017）<ref name="Ince_CCT">{{cite journal|author1=Ince R. A.|title=Measuring multivariate redundant information with pointwise common change in surprisal|journal=Entropy|year=2017|volume=19|issue=7|page=318|url=https://doi.org/10.3390/e19070318}}</ref>基于点对点共同变化不确定度度量冗余信息。Bertschinger等（2014）<ref name="Bertschinger_unique">{{cite journal|author1=Bertschinger N.|author2=Rauh J.|author3=Olbrich E.|author4=Jost J.|author5=Ay N.|title=Quantifying unique information|journal=Entropy|year=2014|volume=16|issue=4|pages=2161–2180|url=https://doi.org/10.3390/e16042161}}</ref>从博弈论/决策论视角定义<math>I_{\cap}^{∧}</math>，通过"能否在不知另一源前提下做出同样决策"来区分特有信息与冗余信息。

James 等（2018）<ref name="James_depcon">{{cite journal|author1=James R. G.|author2=Emenheiser J.|author3=Crutchfield J. P.|title=Unique information via dependency constraints|journal=Journal of Physics A: Mathematical and Theoretical|year=2018|volume=52|issue=1|page=014002|url=https://doi.org/10.1088/1751-8121/aae492}}</ref>将定义冗余信息改为通过依赖约束（dependency constraints）定义特有信息<math>I_{\cap}^{\text{dep}}</math>，要求特有信息分量在边际独立性约束下最大化，给出与<math>I_{\cap}^{∧}</math>不同的另一条路线。Finn等（2018）<ref name="Finn_spec_ambig">{{cite journal|author1=Finn C.|author2=Lizier J. T.|title=Pointwise partial information decomposition using the specificity and ambiguity lattices|journal=Entropy|year=2018|volume=20|issue=4|page=297|url=https://doi.org/10.3390/e20040297}}</ref>将点对点互信息拆分成了正的信息分量（特异性）负的信息分量（模糊性）。与此对应，冗余晶格拆分为了特异性格和模糊性格并据此定义了点对点冗余特异性和点对点冗余模糊性。随后，有两种方式组合这两个定义，一种是直接求期望，一种是从点对点特异性和模糊性中减去点对点特有性、冗余性和协同性，得到点对点特有、冗余和协同信息。Ince（2017）<ref name="Ince_CCT">{{cite journal|author1=Ince R. A.|title=Measuring multivariate redundant information with pointwise common change in surprisal|journal=Entropy|year=2017|volume=19|issue=7|page=318|url=https://doi.org/10.3390/e19070318}}</ref>基于点对点共同变化不确定度度量冗余信息。Bertschinger等（2014）<ref name="Bertschinger_unique">{{cite journal|author1=Bertschinger N.|author2=Rauh J.|author3=Olbrich E.|author4=Jost J.|author5=Ay N.|title=Quantifying unique information|journal=Entropy|year=2014|volume=16|issue=4|pages=2161–2180|url=https://doi.org/10.3390/e16042161}}</ref>从博弈论/决策论视角定义<math>I_{\cap}^{∧}</math>，通过"能否在不知另一源前提下做出同样决策"来区分特有信息与冗余信息。

===不同方法的比较===

===不同方法的比较===

我们将几种 PID 度量方式进行比较：Lyu–Clark–Raviv 的 <math> I_{\cap}^{\text{Exp}} </math><ref name="Lyu_do_PID">{{cite conference|author1=Lyu A.|author2=Clark A.|author3=Raviv N.|title=Explicit formula for partial information decomposition|booktitle=2024 IEEE International Symposium on Information Theory (ISIT)|publisher=IEEE|year=2024|pages=2329–2334|url=https://doi.org/10.1109/ISIT57864.2024.10619369}}</ref>；由 Kolchinsky 提出的 <math>I_{\cap}^{\star}</math><ref name="Kolchinsky_Blackwell">{{cite journal|author1=Kolchinsky A.|title=A novel approach to the partial information decomposition|journal=Entropy|year=2022|volume=24|issue=3|page=403|url=https://doi.org/10.3390/e24030403}}</ref>；由 Williams 和 Beer 提出的 <math>I_{\cap}^{\text{WB}}</math><ref name="Paul">{{cite journal|author1=Paul L. Williams|author2=Randall D. Beer|title=Nonnegative Decomposition of Multivariate Information|journal=arXiv preprint|year=2010|url=https://arxiv.org/abs/1004.2515}}</ref>；由 Barrett 提出的 <math>I_{\cap}^{\text{MMI}}</math><ref name="Paul">{{cite journal|author1=Paul L. Williams|author2=Randall D. Beer|title=Nonnegative Decomposition of Multivariate Information|journal=arXiv preprint|year=2010|url=https://arxiv.org/abs/1004.2515}}</ref>；由 Griffith 等人提出的<math> I_{\cap}^{∧}</math><ref name="Griffith_intersection_CR">{{cite journal|author1=Griffith V.|author2=Chong E. K.|author3=James R. G.|author4=Ellison C. J.|author5=Crutchfield J. P.|title=Intersection information based on common randomness|journal=Entropy|year=2014|volume=16|issue=4|pages=1985–2000|url=https://doi.org/10.3390/e16041985}}</ref>；由 Ince 提出的 <math>I_{\cap}^{\text{Ince}}</math><ref name="Ince_CCT">{{cite journal|author1=Ince R. A.|title=Measuring multivariate redundant information with pointwise common change in surprisal|journal=Entropy|year=2017|volume=19|issue=7|page=318|url=https://doi.org/10.3390/e19070318}}</ref>；由 Finn 和 Lizier 提出的 <math>I_{\cap}^{\text{FL}}</math><ref name="Finn_spec_ambig">{{cite journal|author1=Finn C.|author2=Lizier J. T.|title=Pointwise partial information decomposition using the specificity and ambiguity lattices|journal=Entropy|year=2018|volume=20|issue=4|page=297|url=https://doi.org/10.3390/e20040297}}</ref>；由 Bertschinger 等人提出的<math> I_{\cap}^{\text{BROIA}}</math><ref name="Bertschinger_unique">{{cite journal|author1=Bertschinger N.|author2=Rauh J.|author3=Olbrich E.|author4=Jost J.|author5=Ay N.|title=Quantifying unique information|journal=Entropy|year=2014|volume=16|issue=4|pages=2161–2180|url=https://doi.org/10.3390/e16042161}}</ref>；由 Harder 等人提出的 <math>I_{\cap}^{\text{Harder}}</math><ref name="Harder_IRed">{{cite journal|author1=Harder M.|author2=Salge C.|author3=Polani D.|title=Bivariate measure of redundant information|journal=Physical Review E|year=2013|volume=87|issue=1|page=012130|url=https://doi.org/10.1103/PhysRevE.87.012130}}</ref>；由 James 等人提出的 <math>I_{\cap}^{\text{dep}}</math><ref name="James_depcon">{{cite journal|author1=James R. G.|author2=Emenheiser J.|author3=Crutchfield J. P.|title=Unique information via dependency constraints|journal=Journal of Physics A: Mathematical and Theoretical|year=2018|volume=52|issue=1|page=014002|url=https://doi.org/10.1088/1751-8121/aae492}}</ref>。

我们将几种 PID 度量方式进行比较<ref name="Lyu_do_PID">{{cite conference|author1=Lyu A.|author2=Clark A.|author3=Raviv N.|title=Explicit formula for partial information decomposition|booktitle=2024 IEEE International Symposium on Information Theory (ISIT)|publisher=IEEE|year=2024|pages=2329–2334|url=https://doi.org/10.1109/ISIT57864.2024.10619369}}</ref>：Lyu–Clark–Raviv 的 <math> I_{\cap}^{\text{Exp}} </math><ref name="Lyu_do_PID">{{cite conference|author1=Lyu A.|author2=Clark A.|author3=Raviv N.|title=Explicit formula for partial information decomposition|booktitle=2024 IEEE International Symposium on Information Theory (ISIT)|publisher=IEEE|year=2024|pages=2329–2334|url=https://doi.org/10.1109/ISIT57864.2024.10619369}}</ref>；由 Kolchinsky 提出的 <math>I_{\cap}^{\star}</math><ref name="Kolchinsky_Blackwell">{{cite journal|author1=Kolchinsky A.|title=A novel approach to the partial information decomposition|journal=Entropy|year=2022|volume=24|issue=3|page=403|url=https://doi.org/10.3390/e24030403}}</ref>；由 Williams 和 Beer 提出的 <math>I_{\cap}^{\text{WB}}</math><ref name="Paul">{{cite journal|author1=Paul L. Williams|author2=Randall D. Beer|title=Nonnegative Decomposition of Multivariate Information|journal=arXiv preprint|year=2010|url=https://arxiv.org/abs/1004.2515}}</ref>；由 Barrett 提出的 <math>I_{\cap}^{\text{MMI}}</math><ref name="Paul">{{cite journal|author1=Paul L. Williams|author2=Randall D. Beer|title=Nonnegative Decomposition of Multivariate Information|journal=arXiv preprint|year=2010|url=https://arxiv.org/abs/1004.2515}}</ref>；由 Griffith 等人提出的<math> I_{\cap}^{∧}</math><ref name="Griffith_intersection_CR">{{cite journal|author1=Griffith V.|author2=Chong E. K.|author3=James R. G.|author4=Ellison C. J.|author5=Crutchfield J. P.|title=Intersection information based on common randomness|journal=Entropy|year=2014|volume=16|issue=4|pages=1985–2000|url=https://doi.org/10.3390/e16041985}}</ref>；由 Ince 提出的 <math>I_{\cap}^{\text{Ince}}</math><ref name="Ince_CCT">{{cite journal|author1=Ince R. A.|title=Measuring multivariate redundant information with pointwise common change in surprisal|journal=Entropy|year=2017|volume=19|issue=7|page=318|url=https://doi.org/10.3390/e19070318}}</ref>；由 Finn 和 Lizier 提出的 <math>I_{\cap}^{\text{FL}}</math><ref name="Finn_spec_ambig">{{cite journal|author1=Finn C.|author2=Lizier J. T.|title=Pointwise partial information decomposition using the specificity and ambiguity lattices|journal=Entropy|year=2018|volume=20|issue=4|page=297|url=https://doi.org/10.3390/e20040297}}</ref>；由 Bertschinger 等人提出的<math> I_{\cap}^{\text{BROIA}}</math><ref name="Bertschinger_unique">{{cite journal|author1=Bertschinger N.|author2=Rauh J.|author3=Olbrich E.|author4=Jost J.|author5=Ay N.|title=Quantifying unique information|journal=Entropy|year=2014|volume=16|issue=4|pages=2161–2180|url=https://doi.org/10.3390/e16042161}}</ref>；由 Harder 等人提出的 <math>I_{\cap}^{\text{Harder}}</math><ref name="Harder_IRed">{{cite journal|author1=Harder M.|author2=Salge C.|author3=Polani D.|title=Bivariate measure of redundant information|journal=Physical Review E|year=2013|volume=87|issue=1|page=012130|url=https://doi.org/10.1103/PhysRevE.87.012130}}</ref>；由 James 等人提出的 <math>I_{\cap}^{\text{dep}}</math><ref name="James_depcon">{{cite journal|author1=James R. G.|author2=Emenheiser J.|author3=Crutchfield J. P.|title=Unique information via dependency constraints|journal=Journal of Physics A: Mathematical and Theoretical|year=2018|volume=52|issue=1|page=014002|url=https://doi.org/10.1088/1751-8121/aae492}}</ref>。

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| <math>强独立恒等性</math> || || || || || || || <math>\checkmark</math> ||<math>\checkmark</math>|| <math>\checkmark</math> ||<math>\checkmark</math>

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== PID框架的局限 ==

== PID框架的局限 ==