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==Normalization==

==Normalization==

Clearly, the magnitude of EI (Effective Information) is related to the size of the state space, which poses challenges when comparing [[Markov Chains]] of different scales. To address this issue, we need a [[Causal Measure]] that is as independent of scale effects as possible. Therefore, we normalize EI to derive a metric that is independent of the system size.

Clearly, the magnitude of EI (Effective Information) is related to the size of the state space, which poses challenges when comparing [[Markov Chains]] of different scales. To address this issue, we need a [[Measure of Causal Effect]] that is as independent of scale effects as possible. Therefore, we normalize EI to derive a metric that is independent of the system size.

According to the work of [[Erik Hoel]] and [[Tononi]], the normalization process involves using the entropy under a [[Uniform Distribution]] (i.e., [[Maximum Entropy]]) as the denominator - <math>\log N</math>is used as the denominator to normalize EI, where [math]N[/math] is the number of states <ref name=hoel_2013 /> in the state space [math]\mathcal{X}[/math]. Thus, the normalized EI becomes:

According to the work of [[Erik Hoel]] and [[Tononi]], the normalization process involves using the entropy under a [[Uniform Distribution]] (i.e., [[Maximum Entropy]]) as the denominator - <math>\log N</math>is used as the denominator to normalize EI, where [math]N[/math] is the number of states <ref name=hoel_2013 /> in the state space [math]\mathcal{X}[/math]. Thus, the normalized EI becomes: