更改

</math>

</math>

其中<math>\epsilon</math>和<math>\delta</math>分别表示观测噪音和干预噪音的大小。-->This kind of derivation was first seen in Hoel's 2013 paper [1] and was further discussed in detail in the "Neural Information Squeezer" paper [2].

Among them,<math>\epsilon</math> and <math>\delta</math> represent the magnitude of observation noise and intervention noise, respectively.-->This kind of derivation was first seen in Hoel's 2013 paper [1] and was further discussed in detail in the "Neural Information Squeezer" paper [2].

===High-Dimensional Case===

===High-Dimensional Case===

We can extend the EI (Effective Information) calculation for one-dimensional variables to a more general n-dimensional scenario. Specifically:{{NumBlk|:|

We can extend the EI (Effective Information) calculation for one-dimensional variables to a more general n-dimensional scenario. Specifically:{{NumBlk|:|

This property is referred to as [[Dynamical Reversibility]]. Therefore, in a certain sense, EI measures a form of [[Dynamical Reversibility]] in the Markov chain.

This property is referred to as [[Dynamical Reversibility]]. Therefore, in a certain sense, EI measures a form of [[Dynamical Reversibility]] in the Markov chain.

It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.

It's important to note that the [[Dynamical Reversibility]] of a Markov chain described here is different from the commonly understood [[the reversibility of Markov chains]] in the literature. Dynamical reversibility refers to the invertibility of the Markov transition matrix itself, meaning the operations on each deterministic state in the state space are reversible. However, conventional reversible Markov chains do not require the transition matrix to be invertible but rather exhibit time-reversal symmetry with respect to the steady-state distribution. This symmetry implies that the state distribution sequence formed under the evolution of the dynamics 𝑃 remains the same in both forward and reverse time.