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For these three Markov chains, the state space is X={1,2,3,4}, so the size of their TPM is 4×4.

 

For these three [[Markov Chains]], the state space is [math]\mathcal{X}=\{1,2,3,4\}[/math], so the size of their TPM is [math]4\times 4[/math].

 

==EI of Markov Chains==

==EI of Markov Chains==

In a Markov chain, the state variable at any time Xt​ can be considered as the cause, and the state variable at the next time Xt+1​ can be considered as the effect. Thus, the state transition matrix of a Markov chain is its causal mechanism. Therefore, we can apply the definition of Effective Information to Markov chains.

In a [[Markov Chain]], the state variable at any time [math]X_t[/math] can be considered as the cause, and the state variable at the next time [math]X_{t+1}[/math] can be considered as the effect. Thus, the [[Transitional Probability Matrix]] of a [[Markov Chain]] is its [[Causal Mechanism]]. Therefore, we can apply the definition of Effective Information to the [[Markov Chain]].

 

 

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<math>

\begin{aligned}

\begin{aligned}

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</math>

Here, X~t​,X~t+1​ are the states at times t and t+1 after intervening to make Xt​ uniformly distributed, and pij​ is the probability of transitioning from state i to state j. From this equation, it is clear that EI is merely a function of the probability transition matrix P.

Here, <math>\tilde{X}_t,\tilde{X}_{t+1}</math> are the states at times t and t+1 after [[intervening]] to make [math]X_t[/math] [[Uniformly Distributed]], and <math>p_{ij}</math> is the probability of transitioning from state i to state j. From this equation, it is clear that EI is merely a function of the probability transition matrix [math]P[/math].

 

==Vector Form of EI in Markov Chains==

==Vector Form of EI in Markov Chains==

We can also represent the transition probability matrix P as a concatenation of N row vectors, i.e.:  

We can also represent the transition probability matrix P as a concatenation of N row vectors, i.e.: