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

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## 数学代写|信息论代写information theory代考|Logical Information as the Measure of Distinctions

There is now a widespread view that information is fundamentally about differences, distinguishability, distinctions, and diversity. As Charles H. Bennett, one of the founders of quantum information theory, put it:

So information really is a very useful abstraction. It is the notion of distinguishability abstracted away from what we are distinguishing, or from the carrier of information. [3, p. 155$]$
This view even has an interesting history. In James Gleick’s book, The Information: A History, A Theory, A Flood, he noted the focus on differences in the seventeenth century polymath, John Wilkins, who was a founder of the Royal Society. In 1641 , the year before Isaac Newton was born, Wilkins published one of the earliest books on cryptography, Mercury or the Secret and Swift Messenger, which not only pointed out the fundamental role of differences but noted that any (finite) set of different things could be encoded by words in a binary code.
For in the general we must note, That whatever is capable of a competent Difference, perceptible to any Sense, may be a sufficient Means whereby to express the Cogitations. It is more convenient, indeed, that these Differences should be of as great Variety as the Letters of the Alphabet; but it is sufficient if they be but twofold, because Two alone may, with somewhat more Labour and Time, be well enough contrived to express all the rest. [27, Chap. XVII, p. 69]
Wilkins explains that a five letter binary code would be sufficient to code the letters of the alphabet since $2^{5}=32$.
Thus any two Letters or Numbers, suppose $A . B$. being transposed through five Places, will yield Thirty Two Differences, and so consequently will superabundantly serve for the Four and twenty Letters…. [27, Chap. XVII, p. 69]
As Gleick noted:
Any difference meant a binary choice. Any binary choice began the expressing of cogitations. Here, in this arcane and anonymous treatise of 1641 , the essential idea of information theory poked to the surface of human thought, saw its shadow, and disappeared again for [three] hundred years. [11, p. 161]
Thus counting distinctions would seem the right way to measure information, ${ }^{2}$ and that is the measure which emerges naturally out of partition logic-just as finite logical probability emerges naturally as the measure of counting elements in Boole’s subset logic.

In addition to the philosophy of information literature [2], there is a whole subindustry in mathematics concerned with different so-called “measures” of “entropy” or ‘information’ $[1,26]$ that is long on formulas and ‘intuitive axioms’ but short on interpretations and short on actually being measures. Out of that plethora of definitions, logical entropy is the measure (in the non-negative technical sense of measure theory) of information that arises out of partition logic just as logical probability theory arises out of subset logic.

## 数学代写|信息论代写information theory代考|Classical Logical Probability and Logical Entropy

A partition on a set $U$ is a set $\pi=\left{B, B^{\prime}, \ldots\right}$ of subsets of $U$, called the “blocks” of $\pi$, that are jointly exhaustive (union is $U$ ) and mutually exclusive (disjoint). A distinction or dit of a partition $\pi$ is an ordered pair $\left(u, u^{\prime}\right)$ of elements of $U$ that are in different blocks of $\pi$. The ditset dit $(\pi)$ is the subset of $U \times U$ consisting of all the distinctions of $\pi$. A binary relation $R \subseteq U \times U$ on $U$ is said to be a partition relation (also apartness relation) is it is anti-reflexive, i.e., no self-pairs (u,u) are in $R$, symmetric, i.e., $\left(u, u^{\prime}\right) \in R$ implies $\left(u^{\prime}, u\right) \in R$, and intransitive in the sense that if $\left(u, u^{\prime}\right) \in R$, then for any sequence $u=u_{0}, u_{1}, u_{2}, \ldots, u_{n}, u_{n+1}=u^{\prime}$ of elements of $U$, at least one of the pairs $\left(u_{i}, u_{i+1}\right)$ for $i=0, \ldots, n$ must be in $R$. All ditsets are partition relations and all partition relations are the ditset of some partition on $U$. An indistinction or indit of $\pi$ is the set of ordered pairs $\left(u, u^{\prime}\right)$ of elements in the same block of $\pi$, so the inditset indit $(\pi)=\bigcup_{B \in \pi} B \times B$ is set of indits of $\pi$. The inditset of a partition is the equivalence relation associated with $\pi$, i.e., the equivalence relation whose equivalence classes are blocks of $\pi$. Equivalence relations and partition relations are complementary subsets of $U \times U$.

George Boole [5] extended his logic of subsets to finite logical probability theory where, in the equiprobable case, the probability of a subset $S$ (event) of a finite universe set (outcome set or sample space) $U=\left{u_{1}, \ldots, u_{n}\right}$ was the number of elements in $S$ over the total number of elements: $\operatorname{Pr}(S)=\frac{|S|}{|U|}=\sum_{u j \in S} \frac{1}{\mid U}$. Pierre-Simon Laplace’s classical finite probability theory [18] also dealt with the case where the outcomes were assigned real point probabilities $p=\left{p_{1}, \ldots, p_{n}\right}$ so rather than summing the equal probabilities $\frac{1}{|U|}$, the point probabilities of the elements were summed: $\operatorname{Pr}(S)=\sum_{u_{j} \in S} p_{j}=p(S)$-where the equiprobable

formula is for $p_{j}=\frac{1}{|U|}$ for $j=1, \ldots, n$. The conditional probability of an event $T \subseteq U$ given an event $S$ is $\operatorname{Pr}(T \mid S)=\frac{p(T \cap S)}{p(S)}$.

In Gian-Carlo Rota’s Fubini Lectures [23] (and in his lectures at MIT), he has remarked in view of duality between partitions and subsets that, quantitatively, the “lattice of partitions plays for information the role that the Boolean algebra of subsets plays for size or probability” $[17$, p. 30$]$ or symbolically:
$$\frac{\text { information }}{\text { partitions }} \approx \frac{\text { probability }}{\text { subsets. }} \text {. }$$

## 数学代写|信息论代写information theory代考|Information Algebras and Joint Distributions

Consider a joint probability distribution ${p(x, y)}$ on the finite sample space $X \times Y$ (where to avoid trivialities, assume $|X|,|Y| \geq 2$ ), with the marginal distributions ${p(x)}$ and ${p(y)}$ where $p(x)=\sum_{y \in Y} p(x, y)$ and $p(y)=\sum_{x \in X} p(x, y)$. For notational simplicity, the entropies can be considered as functions of the random variables or of their probability distributions, e.g., $h({p(x, y)})=h(X, Y)$, $h({p(x)})=h(X)$, and $h({p(y)})=h(Y)$. For the joint distribution, we have the:
\begin{aligned} h(X, Y)=& \sum_{x \in X, y \in Y} p(x, y)[1-p(x, y)]=1-\sum_{x, y} p(x, y)^{2} \ & \text { Logical entropy of the joint distribution } \end{aligned}
which is the probability that two samplings of the joint distribution will yield a pair of distinct ordered pairs $(x, y),\left(x^{\prime}, y^{\prime}\right) \in X \times Y$, i.e., with an $X$-distinction $x \neq x^{\prime}$ or a $Y$-distinction $y \neq y^{\prime}$ (since ordered pairs are distinct if distinct on one or more of the coordinates). The logical entropy notions for the probability distribution ${p(x, y)}$ on $X \times Y$ are all product probability measures $\mu(S)$ of certain subsets $S \subseteq(X \times Y)^{2}$ where:
$$\mu(S)=\sum\left{p(x, y) p\left(x^{\prime}, y^{\prime}\right):\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right) \in S\right} .$$
These information sets or infosets are defined solely in terms of equations and inequations (the ‘calculus of identity and difference’) independent of any probability distributions.
For the logical entropies defined so far, the infosets are:
$$\begin{gathered} S_{X}=\left{\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right): x \neq x^{\prime}\right}, \ h(X)=\mu\left(S_{X}\right)=1-\sum_{x} p(x)^{2} ; \ S_{Y}=\left{\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right): y \neq y^{\prime}\right}, \ h(Y)=\mu\left(S_{Y}\right)=1-\sum_{y} p(y)^{2} ; \text { and } \end{gathered}$$

## 数学代写|信息论代写information theory代考|Classical Logical Probability and Logical Entropy

George Boole [5] 将他的子集逻辑扩展到有限逻辑概率论，其中，在等概率的情况下，子集的概率小号有限宇宙集（结果集或样本空间）的（事件）U=\left{u_{1}, \ldots, u_{n}\right}U=\left{u_{1}, \ldots, u_{n}\right}是元素的数量小号在元素总数上：公关⁡(小号)=|小号||在|=∑在j∈小号1∣在. Pierre-Simon Laplace 的经典有限概率理论 [18] 也处理了将结果分配给实点概率的情况p=\left{p_{1}, \ldots, p_{n}\right}p=\left{p_{1}, \ldots, p_{n}\right}所以而不是将相等的概率相加1|在|，元素的点概率相加：公关⁡(小号)=∑在j∈小号pj=p(小号)- 等概率的地方

信息  分区 ≈ 可能性  子集。 .

## 数学代写|信息论代写information theory代考|Information Algebras and Joint Distributions

H(X,是)=∑X∈X,是∈是p(X,是)[1−p(X,是)]=1−∑X,是p(X,是)2  联合分布的逻辑熵

\mu(S)=\sum\left{p(x, y) p\left(x^{\prime}, y^{\prime}\right):\left((x, y),\left( x^{\prime}, y^{\prime}\right)\right) \in S\right} 。\mu(S)=\sum\left{p(x, y) p\left(x^{\prime}, y^{\prime}\right):\left((x, y),\left( x^{\prime}, y^{\prime}\right)\right) \in S\right} 。

\begin{聚集} S_{X}=\left{\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right): x \neq x^ {\prime}\right}, \h(X)=\mu\left(S_{X}\right)=1-\sum_{x} p(x)^{2} ; \ S_{Y}=\left{\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right): y \neq y^{\prime} \right}, \h(Y)=\mu\left(S_{Y}\right)=1-\sum_{y} p(y)^{2} ; \text { 和 } \end{聚集}\begin{聚集} S_{X}=\left{\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right): x \neq x^ {\prime}\right}, \h(X)=\mu\left(S_{X}\right)=1-\sum_{x} p(x)^{2} ; \ S_{Y}=\left{\left((x, y),\left(x^{\prime}, y^{\prime}\right)\right): y \neq y^{\prime} \right}, \h(Y)=\mu\left(S_{Y}\right)=1-\sum_{y} p(y)^{2} ; \text { 和 } \end{聚集}

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