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=Effective Information of Markov Chains=

=Effective Information of Markov Chains=

==Introduction to Markov Chains==

==Introduction to Markov Chains==

In this section, all Markov transition probability matrices are denoted as P. N is the total number of states.

<blockquote>In this section, all Markov transition probability matrices are denoted as [math]P[/math]. N is the total number of states.

Erik Hoel and colleagues first proposed the measure of causality, Effective Information (EI), on the Markov dynamics with discrete states, also known as Markov chains. Therefore, this section introduces the specific form of EI on Markov chains.

[[Erik Hoel]] and colleagues first proposed the measure of causality, Effective Information (EI), on the [[Markov Dynamics]] with discrete states, also known as [[Markov Chains]]. Therefore, this section introduces the specific form of EI on [[Markov Chains]].

A Markov chain refers to a type of stationary stochastic process with discrete states and discrete time. Its dynamics can generally be represented by a Transitional Probability Matrix (TPM), also known as a probability transition matrix or state transition matrix.

A [[Markov Chain]] refers to a type of [[Stationary Stochastic Process]] with discrete states and discrete time. Its dynamics can generally be represented by a [[Transitional Probability Matrix]] (TPM), also known as a [[Probability Transition Matrix]] or [[State Transition Matrix]].

Specifically, a Markov chain consists of a set of random variables Xt​, which take values in the state space X={1,2,⋯,N}, where t typically represents time. A transition probability matrix is a probability matrix, where the element in the i-th row and j-th column, pij​, represents the probability of the system transitioning from state i at any time t to state j at time t+1. Each row satisfies the normalization condition: ∑j=1N​pij​=1.

Specifically, a [[Markov Chain]] consists of a set of random variables [math]X_t[/math], which take values in the state space [math]\mathcal{X}=\{1,2,\cdots,N\}[/math], where [math]t[/math] typically represents time. A [[Transition Probability Matrix]] is a probability matrix, where the element in the [math]i[/math]-th row and [math]j[/math]-th column, [math]p_{ij}[/math], represents the probability of the system transitioning from state [math]i[/math] at any time [math]t[/math] to state [math]j[/math] at time [math]t+1[/math]. Each row satisfies the normalization condition:  

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