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第690行： − 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].+ 第1,035行：
第1,035行： − − 需要注意的是，这里所说的马尔科夫链的[[动力学可逆性]]与通常意义下的[[马尔科夫链的可逆性]]是不等同的。前者的可逆性体现为马尔科夫概率转移矩阵的可逆性，也就是它针对状态空间中的每一个确定性状态的运算都是可逆的，所以也称其为动力学可逆的。但是，文献中通常意义下的可逆的马尔科夫链并不要求转移矩阵是可逆的，而是要以稳态分布为时间反演对称轴，使得在动力学P作用构成的演化下的正向时间形成的状态分布序列和逆向状态分布序列完全相同。
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</math>
</math>
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.