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→Formal Definition
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Here, [math]do(X\sim U(\mathcal{X}))[/math] represents applying a [[do-operator]] on [math]X[/math], making it follow a uniform distribution [math]U(\mathcal{X})[/math] over [math]\mathcal{X}[/math], which corresponds to a [[Maximum Entropy Distribution]]. [math]\tilde{X}[/math] and [math]\tilde{Y}[/math] represent the variables after the [math]do[/math]-intervention on [math]X[/math] and [math]Y[/math], respectively, where:
Here, [math]do(X\sim U(\mathcal{X}))[/math] represents applying a [[do-operator]] on [math]X[/math], making it following a uniform distribution [math]U(\mathcal{X})[/math] over [math]\mathcal{X}[/math], which corresponds to a [[Maximum Entropy Distribution]]. [math]\tilde{X}[/math] and [math]\tilde{Y}[/math] represent the variables after the [math]do[/math]-intervention on [math]X[/math] and [math]Y[/math], respectively, where:
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This means that the main difference between [math]\tilde{X}[/math] after the intervention and [math]X[/math] before the intervention is their distributions: [math]\tilde{X}[/math] follows a uniform distribution over [math]\mathcal{X}[/math], while [math]X[/math] may follow any arbitrary distribution. [math]\#(\mathcal{X})[/math] represents the cardinality of the set [math]\mathcal{X}[/math], or the number of elements in the set if it is finite.
This means that the main difference between [math]\tilde{X}[/math] after the intervention and [math]X[/math] before the intervention is their distributions: [math]\tilde{X}[/math] follows a uniform distribution over [math]\mathcal{X}[/math], while [math]X[/math] may follow any arbitrary distribution. [math]\#(\mathcal{X})[/math] represents the cardinality of the set [math]\mathcal{X}[/math], or the number of elements in the set if it is finite.
According to [[Judea Pearl]]'s theory, the do-operator cuts off all causal arrows pointing to variable [math]X[/math], while keeping other factors unchanged, particularly the causal mechanism from [math]X[/math] to [math]Y[/math]. The [[Causal Mechanism]] is defined as the conditional probability of [math]Y[/math] taking any value [math]\mathcal{Y}[/math] given [math]X[/math] takes a value [math]y\in \mathcal{Y}[/math]:
According to [[Judea Pearl]]'s theory, the do-operator cuts off all causal arrows pointing to variable [math]X[/math], while keeping other factors unchanged, particularly the causal mechanism from [math]X[/math] to [math]Y[/math]. The [[Causal Mechanism]] is defined as the conditional probability of [math]Y[/math] taking any value [math]y\in \mathcal{Y}[/math] given [math]X[/math] takes a value [math]x\in \mathcal{X}[/math]:
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In the intervention, this [[Causal Mechanism]] [math]f[/math] remains constant. When no other variables are influencing the system, this leads to a change in the distribution of [math]Y[/math], which is indirectly intervened upon and becomes:
In the intervention, this [[Causal Mechanism]] [math]f[/math] remains unchanged. When no other variables are influencing the system, this leads to a change in the distribution of [math]Y[/math], which is indirectly intervened upon and becomes:
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Among them, [math]\tilde{Y}[/math] represents the [math]Y[/math] variable indirectly changed by [math]X[/math]'s do-intervention while maintaining the causal mechanism [math]f[/math] unchanged, and this change is mainly reflected in the change of probability distribution.
Among them, [math]\tilde{Y}[/math] represents the [math]Y[/math] variable indirectly changed by [math]X[/math]'s do-intervention while maintaining the causal mechanism [math]f[/math] unchanged, and this change is mainly reflected in the change of probability distribution of [math]Y[/math].
Therefore, the effective information (EI) of a causal mechanism [math]f[/math] is the [[Mutual Information]] between the intervened cause variable [math]\tilde{X}[/math] and the intervened effect variable [math]\tilde{Y}[/math].
Therefore, the effective information (EI) of a causal mechanism [math]f[/math] is the [[Mutual Information]] between the intervened cause variable [math]\tilde{X}[/math] and the intervened effect variable [math]\tilde{Y}[/math].