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==Effective Information as the Distribution Difference==

==Effective Information as the Distribution Difference==

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].

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 effective 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]] between two probability distributions: the prior distribution of the dependent variable [math]\tilde{X}[/math], i.e. the uniform distribution [math]U[/math] on [math]\mathcal{X}[/math], and the posterior distribution of it, i.e., [math]P(\tilde{X}|\tilde{Y}=Y_0,f)[/math], which is the inferred distribution of [math]\tilde{X}[/math] after the event [math]\tilde{Y}=Y_0[/math] is observed. And this event is the effect caused by the intervention of [math]do(X\sim U(\mathcal{X}))[/math] under the causal mechanism [math]f[/math].

So, there will be a difference between the prior probability distribution and the posterior probability distribution, which is the effective information generated by the causal mechanism f, and can be defined as:

So, there will be a difference between the prior probability distribution and the posterior probability distribution, which is the effective information generated by the causal mechanism f, and can be defined as:

</math>

</math>

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 (Integrated Information Theory 3.0 Version<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]].

Here, [math]\tilde{X}[/math] and [math]\tilde{Y}[/math] respectively represent the cause and effect 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 (Integrated Information Theory 3.0 Version<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 [[Earth Moving 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 equation {{EquationNote|1}}. To understand this, we first need to introduce the [[Bayesian Formula]], which is:

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 equation {{EquationNote|1}}. To understand this, we first need to introduce the [[Bayesian Formula]], which is:

</math>

</math>

The introduction of [math]ei[/math] helps us understand how a local causal mechanism changes the distribution of the original variable, or in [[Tononi]]'s language, it is a mechanism of information generation, as detailed in the article<ref name=tononi_2008 /> or the [[Integrated Information Theory]].

The introduction of [math]ei[/math] helps us to understand how a local causal mechanism changes the distribution of the original variable, or in [[Tononi]]'s language, it is a mechanism of information generation, as detailed in the article<ref name=tononi_2008 /> or the [[Integrated Information Theory]].

=Effective Information of Markov Chains=

=Effective Information of Markov Chains=