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==Why Intervene to Achieve a Uniform Distribution?==
==Why Intervene to Achieve a Uniform Distribution?==
In Erik Hoel's original definition, the ''do'' operation intervenes on the dependent variable X, transforming it into a uniform distribution over its domain X (which is also the maximum entropy distribution). So, why should we intervene to achieve a uniform distribution? Can other distributions be used?
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?
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？
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.
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.
The reason for intervening to achieve a uniform distribution for the input variable is to more accurately characterize the properties of the causal mechanism.
The reason for intervening to achieve a uniform distribution for the input variable is to more accurately characterize the properties of the causal mechanism.
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|:|
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。
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。
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：
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|:|
<math>
<math>
\begin{aligned}
\begin{aligned}
</math>
</math>
|{{EquationRef|1}}}}
|{{EquationRef|1}}}}
It is easy to see that the final equation tells us that EI is actually composed of two terms: the first term is the average of the negative entropy of each row of the causal mechanism matrix, and the second term is the entropy of the variable Y. In the first term, the probability distribution Pr(X=x) of X acts as the weight when averaging the entropy of each row. Only when we set this weight to be the same value (i.e., intervene to make X uniformly distributed) can we treat each row of the causal mechanism matrix equally.
It is easy to see that the final equation tells us that EI is actually composed of two terms: the first term is the average of the negative entropy of each row of the causal mechanism matrix, and the second term is the entropy of the variable Y. In the first term, the probability distribution Pr(X=x) of X acts as the weight when averaging the entropy of each row. Only when we set this weight to be the same value (i.e., intervene to make X uniformly distributed) can we treat each row of the causal mechanism matrix equally.