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==Effective Information as the Distribution Difference==

==Effective Information as the Distribution Difference==

在文献<ref name='tononi_2008'>{{cite journal|author=GIULIO TONONI|title=Consciousness as Integrated Information: a Provisional Manifesto|journal=Biol. Bull.|volume=215|page=216–242|year=2008}}</ref>中，作者用另一种方式定义了有效信息。这种新形式的有效信息依赖于果变量（Y）的状态，即干预[math]X[/math]为均匀分布以后的[math]\tilde{Y}[/math]的状态为给定的值[math]Y_0[/math]。在这一条件下，有效信息定义为两种分布的[[KL散度]]，这两种概率分布分别是因变量[math]X[/math]的先验分布，即[math]\mathcal{X}[/math]上的均匀分布[math]U[/math]，以及在从X到Y的因果机制f的作用下，导致果变量[math]Y[/math]变成了另一个变量[math]\tilde{Y}[/math]，那么以观察到这一果变量[math]Y[/math]取值为[math]Y_0[/math]为条件，我们可以反过来推断出因变量[math]\tilde{X}[/math]的后验分布，即[math]P(\tilde{X}|\tilde{Y}=Y_0,f)[/math]。

In the literature<ref name='tononi_2008'>{{cite journal|author=GIULIO TONONI|title=Consciousness as Integrated Information: a Provisional Manifesto|journal=Biol. Bull.|volume=215|page=216–242|year=2008}}</ref>, the author defines valid information in another way. This new form of effective information depends on the state of the outcome variable (Y), that is, the state of [math]\tilde{Y}[/math] after intervening [math]X[/math] to be uniformly distributed is the given value [math]Y_0[/math]. Under this condition, effective information is defined as the [[KL Divergence]] of two probability distributions, which are the prior distributions of the dependent variable [math]X[/math], i.e. the uniform distribution [math]U[/math] on [math]\mathcal{X}[/math], and the causal mechanism f from X to Y, which causes the dependent variable [math]Y[/math] to become another variable [math]\tilde{Y}[/math]. Therefore, based on the observation that the value of this dependent variable [math]Y[/math] is [math]Y_0[/math], we can infer in reverse that The posterior distribution of the dependent variable [math]\tilde{X}[/math], i.e. [math]P(\tilde{X}|\tilde{Y}=Y_0,f)[/math].

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这里，[math]\tilde{X}[/math]和[math]\tilde{Y}[/math]分别表示将[math]X[/math]干预成均匀分布后（即先验分布），在因果机制[math]f[/math]保持不变的前提下的因变量和果变量。值得注意的是，在文献<ref name='tononi_2008'/>中，作者没有明确给出[[KL散度]]的形式，在后续的文献中（整合信息论3.0版本<ref name='IIT3.0'>{{cite journal|author1=Oizumi M|author2=Albantakis L|author3=Tononi G|year=2014|title=From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0|journal=PLoS Computational Biology|volume=10|number=5|page=e1003588}}</ref>），作者使用了其它有关概率分布距离的对称性度量形式，例如[[推土距离]]。

Here, [math]\tilde{X}[/math] and [math]\tilde{Y}[/math] respectively represent the dependent and dependent variables after intervening [math]X[/math] into a uniform distribution (i.e. prior distribution), while keeping the causal mechanism [math]f[/math] unchanged. It is worth noting that in the literature<ref name='tononi_2008'/>, the author did not explicitly provide the form of the [[KL Divergence]]. In subsequent literature (<ref name='IIT3.0'>{{cite journal|author1=Oizumi M|author2=Albantakis L|author3=Tononi G|year=2014|title=From the Phenomenology to the Mechanisms of Consciousness: Integrated Information Theory 3.0|journal=PLoS Computational Biology|volume=10|number=5|page=e1003588}}</ref>), the author used other symmetry measures related to probability distribution distance, such as the [[Bulldozing Distance]].

In fact, [math]ei(f,Y_0)[/math] is the effective information value under a certain [math]Y_0[/math] value. If we take the average of all [math]Y_0[/math] values, we can obtain the effective information in the usual sense, which is the {{EquationNote|1}} equation. To understand this, we first need to introduce the [[Bayesian Formula]], which is:

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这里的[math]\tilde{Y_0}\equiv (\tilde{Y}=Y_0)[/math].

这里的[math]\tilde{Y_0}\equiv (\tilde{Y}=Y_0)[/math].

注意，这里的条件概率[math]P(\tilde{Y_0}|\tilde{X})[/math]事实上就是因果机制[math]f[/math]，进一步，把它代入[math]ei(f,Y_0)[/math]的公式，我们不难得到：

注意，这里的条件概率[math]P(\tilde{Y_0}|\tilde{X})[/math]事实上就是因果机制[math]f[/math]，进一步，把它代入[math]ei(f,Y_0)[/math]的公式，我们不难得到：

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