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第341行：
第341行： − 在离散状态和离散时间的马尔科夫链上，EI的定义域显然是概率转移矩阵P。P是一个由[math]N\times N[/math]个元素构成的矩阵，其中每一个元素[math]p_{ij}\in[0,1][/math]代表一个概率值，同时对于任意的行，这组概率值需要满足归一化条件，也就是对于任意的[math]\forall i\in[1,N][/math]:+ 第364行：
第364行： + 第378行：
第379行： − 进一步地，为了求得EI函数的凸性，我们可以求出EI这个函数的二阶导数，其中<math>1\leq s \leq N, 1\leq t \leq N-1 </math>。首先我们需要引入一个函数符号<math>\delta_{i,j} </math>，+ − − 第390行：
第389行： − 于是我们可以来推导EI的二阶导数，当<math>i=s </math>时，+ 第402行：
第401行： − 当<math>i\ne s</math>时，+ 第410行：
第409行： − 综上，EI的二阶导数为，+ 第418行：
第417行： − 并且，二阶导数在<math>i=s </math>的时候为正，在<math>i\ne s</math>时为负。因此EI既不是凸的也不是凹的。+ − + − 前面已经根据EI的一阶导数讨论了它的极值问题。即EI会在所有行向量都是同一个向量的时候达到极小值0。那么，这个极小值是不是最小的呢？+ − 我们知道，根据公式{{EquationNote|2}}，EI本质上是一种KL散度，而KL散度都是非负的，也就是说：+ 第428行：
第427行： − 这个不等式的等号在[math]p_{ij}=\bar{p}_{\cdot j}[/math]对所有的[math]i,j[/math]都成立的时候能够取到。因此，0是EI的最小值，且[math]p_{ij}=\bar{p}_{\cdot j}=\frac{1}{N}\sum_{k=1}^Np_{kj}[/math]是它的最小值点，即：+ − + − 通过前面确定性和简并性章节的讨论，我们知道EI可以拆成两部分，+ 第440行：
第439行： − 其中熵的定义是<math>H(P_i)=-\sum_{j=1}^Np_{ij}\log p_{ij}</math>。分开来看，前一项式子最大值为0，即：+ + +
无编辑摘要
From Equation 2, we can see that in the transition probability matrix P, EI is a function of each element, representing the conditional probabilities of transitioning from one state to another. Thus, a natural question arises: What mathematical properties does this function have? Does it have extreme points, and if so, where are they? Is it convex? What are its maximum and minimum values?
From Equation 2, we can see that in the transition probability matrix P, EI is a function of each element, representing the conditional probabilities of transitioning from one state to another. Thus, a natural question arises: What mathematical properties does this function have? Does it have extreme points, and if so, where are they? Is it convex? What are its maximum and minimum values?
===Domain===
===Domain===
In the case of Markov chains with discrete states and discrete time, the domain of EI is clearly the transition probability matrix P. P is a matrix composed of N×N elements, each representing a probability value pij∈[0,1]. Additionally, each row must satisfy the normalization condition, meaning for any ∀i∈[1,N], the sum of the row's probabilities equals:{{NumBlk|:|
In the case of Markov chains with discrete states and discrete time, the domain of EI is clearly the transition probability matrix P. P is a matrix composed of N×N elements, each representing a probability value pij∈[0,1]. Additionally, each row must satisfy the normalization condition, meaning for any ∀i∈[1,N], the sum of the row's probabilities equals:{{NumBlk|:|
</math>
</math>
|{{EquationRef|3}}}}
|{{EquationRef|3}}}}
It is straightforward to compute that in this case, EI=0, indicating that EI reaches an extreme point. From the second derivative of EI, we can easily determine that this is a minimum point. In other words, EI has many minimum points, as long as all the row vectors of the transition probability matrix are identical. Regardless of the specific distribution of these row vectors, EI will be zero.
It is straightforward to compute that in this case, EI=0, indicating that EI reaches an extreme point. From the second derivative of EI, we can easily determine that this is a minimum point. In other words, EI has many minimum points, as long as all the row vectors of the transition probability matrix are identical. Regardless of the specific distribution of these row vectors, EI will be zero.
===Second Derivative and Convexity===
===Second Derivative and Convexity===
To explore the convexity of the EI function, we can compute its second derivative<math>\frac{\partial^2 EI}{\partial p_{ij}\partial p_{st}}</math>, where 1≤s≤N and 1≤t≤N−1. First, we introduce a function symbol δi,j,
To explore the convexity of the EI function, we can compute its second derivative<math>\frac{\partial^2 EI}{\partial p_{ij}\partial p_{st}}</math>, where 1≤s≤N and 1≤t≤N−1. First, we introduce a function symbol δi,j, and then proceed to derive the second derivative. When i=s:
<math>
<math>
</math>
</math>
Then, proceed to derive the second derivative. When <math>i=s </math>，
<math>
<math>
</math>
</math>
When <math>i\ne s</math>,
<math>
<math>
</math>
</math>
In summary, the second derivative of EI is:
<math>
<math>
</math>
</math>
Moreover, the second derivative is positive when i=s and negative when i=s. Therefore, EI is neither convex nor concave.
===最小值===
===Minimum Value===
From the first derivative of EI, we discussed the extreme value problem, showing that EI reaches a minimum of 0 when all the row vectors of the transition matrix are identical. Is this minimum value the lowest possible?
We know that based on Equation 2, EI is essentially a form of KL divergence, which is always non-negative. Thus:
<math>
<math>
</math>
</math>
The equality holds when pij=pˉ⋅j for all i,j. Hence, 0 is the minimum value of EI, with the minimum point being:
<math>
<math>
EI_{min}=0.
EI_{min}=0.
</math>
</math>
===最大值===
===Maximum Value===
From the previous discussion on determinism and degeneracy, we know that EI can be decomposed into two parts:
<math>
<math>
</math>
</math>
Where H(Pi) is the entropy of row vector Pi. The maximum value of the first term is 0:<math>H(P_i)=-\sum_{j=1}^Np_{ij}\log p_{ij}</math>。分开来看，前一项式子最大值为0，即：
<math>
<math>
- H(P_i)\leq 0,
- H(P_i)\leq 0,
</math>
</math>
This occurs when Pi is a deterministic one-hot vector. Therefore, when all Pi are one-hot vectors, we have:
当<math>P_i</math>为没有不确定性的独热向量时，该式子的等号成立。所以，当对所有的i都有<math>P_i</math>为独热变量时，有
当<math>P_i</math>为没有不确定性的独热向量时，该式子的等号成立。所以，当对所有的i都有<math>P_i</math>为独热变量时，有