### 数学代写|信息论代写information theory代考|The Relationship Between Logical Entropy and Shannon Entropy

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## 数学代写|信息论代写information theory代考|Brief History of the Logical Entropy Formula

The logical entropy formula $h(p)=\sum_{i} p_{i}\left(1-p_{i}\right)=1-\sum_{i} p_{i}^{2}$ is the probability of getting distinct values $u_{i} \neq u_{j}$ in two independent samplings of the random variable $u$. The complementary measure $1-h(p)=\sum_{i} p_{i}^{2}$ is the probability that the two drawings yield the same value from $U$. Thus $1-\sum_{i} p_{i}^{2}$ is a measure of heterogeneity or diversity in keeping with our theme of information as distinctions, while the complementary measure $\sum_{i} p_{i}^{2}$ is a measure of homogeneity or concentration. Historically, the formula can be found in either form depending on the particular context. The $p_{i}$ ‘s might be relative shares such as the relative share of organisms of the $i$ th species in some population of organisms, and then

the interpretation of $p_{i}$ as a probability arises by considering the random choice of an organism from the population.

According to I. J. Good, the formula has a certain naturalness: “If $p_{1}, \ldots, p_{t}$ are the probabilities of $t$ mutually exclusive and exhaustive events, any statistician of this century who wanted a measure of homogeneity would have take about two seconds to suggest $\sum p_{i}^{2}$ which I shall call $\rho . “[13$, p. 561] As noted by Bhargava and Uppuluri [4], the formula $1-\sum p_{i}^{2}$ was used by Gini in 1912 [10] as a measure of “mutability” or diversity. But another development of the formula (in the complementary form) in the early twentieth century was in cryptography. The American cryptologist, William F. Friedman, devoted a 1922 book [9] to the “index of coincidence” (i.e., $\sum p_{i}^{2}$ ). Solomon Kullback (see the Kullback-Leibler divergence treated later) worked as an assistant to Friedman and wrote a book on cryptology which used the index [16].

During World War II, Alan M. Turing worked for a time in the Government Code and Cypher School at the Bletchley Park facility in England. Probably unaware of the earlier work, Turing used $\rho=\sum p_{i}^{2}$ in his cryptoanalysis work and called it the repeat rate since it is the probability of a repeat in a pair of independent draws from a population with those probabilities (i.e., the identification probability $1-h(p)$ ). Polish cryptoanalysts had independently used the repeat rate in their work on the Enigma [21].

## 数学代写|信息论代写information theory代考|Shannon Entropy

For a partition $\pi=\left{B_{1}, \ldots, B_{m}\right}$ with block probabilities $p\left(B_{i}\right)$ (obtained using equiprobable points or with point probabilities), the Shannon entropy of the partition (using logs to base 2) is:
$$H(\pi)=-\sum_{i=1}^{m} p\left(B_{i}\right) \log \left(p\left(B_{i}\right)\right)$$
Or if given a finite probability distribution $p=\left{p_{1}, \ldots, p_{m}\right}$, the Shannon entropy of the probability distribution is:
$$H(p)=-\sum_{i=1}^{m} p_{i} \log \left(p_{i}\right)$$
The Shannon entropy is often presented as being the same as the Boltzmann entropy $S=\frac{1}{n} \ln \left(\frac{n !}{n_{1} ! \ldots n_{m} !}\right)$; it is even called the “Shannon-Boltzmann entropy” by

many authors. The story is that when presented with Shannon’s formula, John von Neumann suggested calling it “entropy” since it occurs in statistical mechanicsand since Shannon could always win arguments since no one really knows what entropy is. But it only occurs in statistical mechanics as a numerical approximation based on taking the first two terms in the Stirling approximation formula for $\ln (n !)$. The first two terms in the Stirling approximation for $\ln (N !)$ are: $\ln (N !) \approx$ $N \ln (N)-N$

If we consider a partition on a finite $U$ with $|U|=N$, with $n$ blocks of size $N_{1}, \ldots, N_{n}$, then the number of ways of distributing the individuals in these $n$ boxes with those numbers $N_{i}$ in the $i$ th box is: $W=\frac{N !}{N_{1} ! \times \ldots \times N_{n} !}$. The normalized natural $\log$ of $W, S=\frac{1}{N} \ln (W)$ is one form of entropy in statistical mechanics. Indeed, the formula $S=k \log (W)$ is engraved on Boltzmann’s tombstone.

The entropy formula can then be developed using the first two terms in the Stirling approximation.
$$\begin{gathered} S=\frac{1}{N} \ln (W)=\frac{1}{N} \ln \left(\frac{N !}{N_{1} ! \times \ldots \times N_{n} !}\right)=\frac{1}{N}\left[\ln (N !)-\sum_{i} \ln \left(N_{i} !\right)\right] \ \approx \frac{1}{N}\left[N[\ln (N)-1]-\sum_{i} N_{i}\left[\ln \left(N_{i}\right)-1\right]\right] \ =\frac{1}{N}\left[N \ln (N)-\sum N_{i} \ln \left(N_{i}\right)\right]=\frac{1}{N}\left[\sum N_{i} \ln (N)-\sum N_{i} \ln \left(N_{i}\right)\right] \ =\sum \frac{N_{i}}{N} \ln \left(\frac{1}{N_{i} / N}\right)=\sum p_{i} \ln \left(\frac{1}{p_{i}}\right)=H_{e}(p) \end{gathered}$$

## 数学代写|信息论代写information theory代考|Logical Entropy, Not Shannon Entropy

Shannon entropy and the many other suggested ‘entropies’ (Rényi, Tsallis, etc.) are routinely called “measures” of information in the general sense of a real-valued quantification -but not in the sense of measure theory. The formulas for mutual information, joint entropy, and conditional entropy are defined so these Shannon entropies satisfy Venn diagram formulas. As Lorne Campbell put it:
Certain analogies between entropy and measure have been noted by various authors. These analogies provide a convenient mnemonic for the various relations between entropy, conditional entropy, joint entropy, and mutual information. It is interesting to speculate whether these analogies have a deeper foundation. It would seem to be quite significant if entropy did admit an interpretation as the measure of some set. [1, p. 112]

For any finite set $U$, a measure $\mu$ is a function $\mu: \wp(U) \rightarrow \mathbb{R}$ such that:

1. $\mu(\emptyset)=0$,
2. for any $E \subseteq U, \mu(E) \geq 0$, and
3. for any disjoint subsets $E_{1}$ and $E_{2}, \mu\left(E_{1} \cup E_{2}\right)=\mu\left(E_{1}\right)+\mu\left(E_{2}\right)$.
The standard usage in measure theory $[4,3]$ seems to be that a “measure” is defined to be non-negative, and the extension to allow negative values is a “signed measure.” That definition is used here although a few authors [10] define a measure to allow negative values and then call the restriction to non-negative values a “positive measure.” The point to notice is that both measures and signed measures can be represented as Venn diagrams (allowing negative areas). As we will see, for three or more random variables, the Shannon mutual information can have negative values-which has no known interpretation. But there is an interesting difference. The logical entropies are defined in terms of a probability measure on a set. The compound Shannon entropies are defined ‘directly’ so as to satisfy the Venn diagram relationships without any mention of a set on which the Venn diagram is defined. As one author put it: “Shannon carefully contrived for this ‘accident’ to occur” [11, p. 153]. But it is possible ex post to then define a set so that the alreadydefined Shannon entropies are the appropriate values on subsets of that set. The ex post construction for the Shannon entropies was first carried out by Kuo Ting Hu [6] but was also noted by Imre Csiszar and Janos Körner [2], and redeveloped by Raymond Yeung [14]. Outside the context of Shannon entropies, the underlying mathematical facts about additive set functions and the inclusion-exclusion principle were known at least from the 1925 first edition of the Polya-Szego classic [9] and Ryser’s treatment of combinatorics [12]-all well-developed in [13].

## 数学代写|信息论代写information theory代考|Shannon Entropy

H(圆周率)=−∑一世=1米p(乙一世)日志⁡(p(乙一世))

H(p)=−∑一世=1米p一世日志⁡(p一世)

## 数学代写|信息论代写information theory代考|Logical Entropy, Not Shannon Entropy

1. μ(∅)=0,
2. 对于任何和⊆在,μ(和)≥0， 和
3. 对于任何不相交的子集和1和和2,μ(和1∪和2)=μ(和1)+μ(和2).
测度论中的标准用法[4,3]似乎“度量”被定义为非负数，而允许负值的扩展是“有符号度量”。尽管一些作者 [10] 定义了一个允许负值的度量，然后将对非负值的限制称为“正度量”，但这里使用了该定义。需要注意的一点是，度量和签名度量都可以表示为维恩图（允许负区域）。正如我们将看到的，对于三个或更多随机变量，香农互信息可以具有负值——这没有已知的解释。但是有一个有趣的区别。逻辑熵是根据集合上的概率度量来定义的。复合香农熵被“直接”定义以满足维恩图关系，而无需提及定义维恩图的集合。正如一位作者所说：“香农精心设计了这个‘意外’的发生”[11, p. 153]。但是可以事后定义一个集合，以便已经定义的香农熵是该集合子集上的适当值。香农熵的事后构造首先由 Kuo Ting Hu [6] 进行，但 Imre Csiszar 和 Janos Körner [2] 也注意到了这一点，并由 Raymond Yeung [14] 重新开发。在香农熵的范围之外，关于加法集函数和包含-排除原理的基本数学事实至少从 1925 年第一版的 Polya-Szego 经典 [9] 和 Ryser 对组合学的处理 [12] 中得知——一切都很好-在[13]中开发。但是可以事后定义一个集合，以便已经定义的香农熵是该集合子集上的适当值。香农熵的事后构造首先由 Kuo Ting Hu [6] 进行，但 Imre Csiszar 和 Janos Körner [2] 也注意到了这一点，并由 Raymond Yeung [14] 重新开发。在香农熵的范围之外，关于加法集函数和包含-排除原理的基本数学事实至少从 1925 年第一版的 Polya-Szego 经典 [9] 和 Ryser 对组合学的处理 [12] 中得知——一切都很好-在[13]中开发。但是可以事后定义一个集合，以便已经定义的香农熵是该集合子集上的适当值。香农熵的事后构造首先由 Kuo Ting Hu [6] 进行，但 Imre Csiszar 和 Janos Körner [2] 也注意到了这一点，并由 Raymond Yeung [14] 重新开发。在香农熵的范围之外，关于加法集函数和包含-排除原理的基本数学事实至少从 1925 年第一版的 Polya-Szego 经典 [9] 和 Ryser 对组合学的处理 [12] 中得知——一切都很好-在[13]中开发。香农熵的事后构造首先由 Kuo Ting Hu [6] 进行，但 Imre Csiszar 和 Janos Körner [2] 也注意到了这一点，并由 Raymond Yeung [14] 重新开发。在香农熵的范围之外，关于加法集函数和包含-排除原理的基本数学事实至少从 1925 年第一版的 Polya-Szego 经典 [9] 和 Ryser 对组合学的处理 [12] 中得知——一切都很好-在[13]中开发。香农熵的事后构造首先由 Kuo Ting Hu [6] 进行，但 Imre Csiszar 和 Janos Körner [2] 也注意到了这一点，并由 Raymond Yeung [14] 重新开发。在香农熵的范围之外，关于加法集函数和包含-排除原理的基本数学事实至少从 1925 年第一版的 Polya-Szego 经典 [9] 和 Ryser 对组合学的处理 [12] 中得知——一切都很好-在[13]中开发。

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## MATLAB代写

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