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Effective Information (查看源代码)

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==Why Intervene to Achieve a Uniform Distribution?==

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In [[Erik Hoel]]'s original definition, the [[do-operator]] intervenes on the dependent variable [math]X[/math], transforming it into a ~~uniform distribution~~ over its domain [math]\mathcal{X}[/math] (which is also the ~~maximum entropy distribution). So, why should we intervene to achieve a uniform distribution? Can other distributions be used?~~

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In [[Erik Hoel]]'s original definition, the [[do-operator]] intervenes on the dependent variable [math]X[/math], transforming it into a [[Uniform Distribution]] over its domain [math]\mathcal{X}[/math] (which is also the [[Maximum Entropy Distribution]])<ref name=hoel_2013 /><ref name=hoel_2017>{{cite journal|author1=Hoel, E.P.|title=When the Map Is Better Than the Territory|journal=Entropy|year=2017|volume=19|page=188|url=https://doi.org/10.3390/e19050188}}</ref>. So, why should we intervene to achieve a [[Uniform Distribution]]? Can other distributions be used?

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在[[~~Erik Hoel~~]]~~的原始定义中，[[do操作]]是将因变量[math]X[/math]干预成了在其定义域[math]\mathcal{X}[/math]上的[[均匀分布]]（也就是[[最大熵分布]]）~~<ref name=hoel_2013 /><ref name=hoel_2017>{{cite journal|author1=Hoel, E.P.|title=When the Map Is Better Than the Territory|journal=Entropy|year=2017|volume=19|page=188|url=https://doi.org/10.3390/e19050188}}</ref>~~。那么，为什么要干预成~~[[~~均匀分布~~]]~~呢？其它分布是否也可以？~~

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Firstly, according to the previous section, the essence of the ''do~~'' operation~~ is to allow the Effective Information (EI) to better describe the nature of the ~~causal mechanism~~ f. Therefore, it is necessary to sever the connection between the dependent variable X and other variables and change its distribution so that the EI metric becomes independent of the distribution of X.

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Firstly, according to the previous section, the essence of the [[do-operator]] is to allow the Effective Information (EI) to better describe the nature of the [[Causal Mechanism]] [math]f[/math]. Therefore, it is necessary to sever the connection between the dependent variable [math]X[/math] and other variables and change its distribution so that the EI metric becomes independent of the distribution of [math]X[/math].

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The reason for intervening to achieve a ~~uniform distribution~~ for the input variable is to more accurately characterize the properties of the ~~causal mechanism~~.

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The reason for intervening to achieve a [[Uniform Distribution]] for the input variable is to more accurately characterize the properties of the [[Causal Mechanism]].

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This is because, when both [math]\mathcal{X}[/math] and [math]\mathcal{Y}[/math] are finite, countable sets, the causal mechanism [math]f\equiv Pr(Y=y|X=x)[/math] becomes a matrix with [math]\#(\mathcal{X})[/math] rows and [math]\#(\mathcal{Y})[/math] columns. We can expand the definition of EI:

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首先，根据上一小节的论述，[[do操作]]的实质是希望让EI能够更清晰地刻画[[因果机制]][math]f[/math]的性质，因此，需要切断因变量[math]X[/math]与其它变量的联系，并改变其分布，让EI度量与[math]X[/math]的分布无关。

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~~而之所以要把输入变量干预为[[均匀分布]]，其实就是要更好地刻画[[因果机制]]的特性。~~

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这是因为，当[math]\mathcal{X}[/math]和[math]\mathcal{Y}[/math]都是有限可数集合的时候，因果机制[math]f\equiv Pr(Y=y|X=x)[/math]就成为了一个[math]\#(\mathcal{X})[/math]行[math]\#(\mathcal{Y})[/math]列的矩阵，我们可以展开EI的定义：

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~~This is because, when both X and Y are finite, countable sets, the causal mechanism f≡Pr(Y=y∣X=x) becomes a matrix with #(X) rows and #(Y) columns. We can expand the definition of EI:~~

{{NumBlk|:|

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