### 数学代写|信息论代写information theory代考|COMP2610

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|信息论代写information theory代考|Definition of information and entropy in the absence of noise

In modern science, engineering and public life, a big role is played by information and operations associated with it: information reception, information transmission, information processing, storing information and so on. The significance of information has seemingly outgrown the significance of the other important factor, which used to play a dominant role in the previous century, namely, energy.

In the future, in view of a complexification of science, engineering, economics and other fields, the significance of correct control in these areas will grow and, therefore, the importance of information will increase as well.

What is information? Is a theory of information possible? Are there any general laws for information independent of its content that can be quite diverse? Answers to these questions are far from obvious. Information appears to be a more difficult concept to formalize than, say, energy, which has a certain, long established place in physics.

There are two sides of information: quantitative and qualitative. Sometimes it is the total amount of information that is important, while other times it is its quality, its specific content. Besides, a transformation of information from one format into another is technically a more difficult problem than, say, transformation of energy from one form into another. All this complicates the development of information theory and its usage. It is quite possible that the general information theory will not bring any benefit to some practical problems, and they have to be tackled by independent engineering methods.

Nevertheless, general information theoory exists, and so dō standärd situations and problems, in which the laws of general information theory play the main role. Therefore, information theory is important from a practical standpoint, as well as in fundamental science, philosophy and expanding the horizons of a researcher.

From this introduction one can gauge how difficult it was to discover the laws of information theory. In this regard, the most important milestone was the work of Claude Shannon $[44,45]$ published in 1948-1949 (the respective English originals are $[38,39]$ ). His formulation of the problem and results were both perceived as a surprise. However, on closer investigation one can see that the new theory extends and develops former ideas, specifically, the ideas of statistical thermodynamics due to Boltzmann. The deep mathematical similarities between these two directions are not accidental. It is evidenced in the use of the same formulae (for instance, for entropy of a discrete random variable). Besides that, a logarithmic measure for the amount of information, which is fundamental in Shannon’s theory, was proposed for problems of communication as early as 1928 in the work of R. Hartley [19] (the English original is [18]).

In the present chapter, we introduce the logarithmic measure of the amount of information and state a number of important properties of information, which follow from that measure, such as the additivity property.

The notion of the amount of information is closely related to the notion of entropy, which is a measure of uncertainty. Acquisition of information is accompanied by a decrease in uncertainty, so that the amount of information can be measured by the amount of uncertainty or entropy that has disappeared.

In the case of a discrete message, i.e. a discrete random variable, entropy is defined by the Boltzmann formula
$$H_{\xi}=-\sum_{\xi} P(\xi) \ln P(\xi),$$
where $\xi$ is a random variable, and $P(\xi)$ is its probability distribution.

## 数学代写|信息论代写information theory代考|Definition of entropy in the case of equiprobable outcomes

Suppose we have $M$ equiprobable outcomes of an experiment. For example, when we roll a standard die, $M=6$. Of course, we cannot always perform the formalization of conditions so easily and accurately as in the case of a die. We assume though that the formalization has been performed and, indeed, one of $M$ outcomes is realized, and they are equivalent in probabilistic terms. Then there is a priori uncertainty directly connected with $M$ (i.e. the greater the $M$ is, the higher the uncertainty is). The quantity measuring the above uncertainty is called entropy and is denoted by $H:$
$$H=f(M),$$
where $f(\cdot)$ is some increasing non-negative function defined at least for natural numbers.

When rolling a dice and observing the outcome number, we obtain information whose amount is denoted by $I$. After that (i.e. a posteriori) there is no uncertainty left: the a posteriori number of outcomes is $M=1$ and we must have $H_{\mathrm{ps}}=f(1)=0$. It is natural to measure the amount of information received by the value of disappeared uncertainty:
$$I=H_{\mathrm{pr}}-H_{\mathrm{ps}} .$$
Here, the subscript ‘pr’ means ‘a priori’, whereas ‘ps’ means ‘a posteriori’.
We see that the amount of received information $I$ coincides with the initial entropy. In other cases (in particular, for formula (1.2.3) given below) a message having entropy $H$ can also transmit the amount of information $I$ equal to $H$.

In order to determine the form of function $f(\cdot)$ in (1.1.1) we employ very natural additivity principle. In the case of a die it reads: the entropy of two throws of a die is twice as large as the entropy of one throw, the entropy of three throws of a die is three times as large as the entropy of one throw, etc. Applying the additivity principle to other cases means that the entropy of several independent systems is equal to the sum of the entropies of individual systems. However, the number $M$ of outcomes for a complex system is equal to the product of the numbers $m$ of outcomes for each one of the ‘simple’ (relative to the total system) subsystems. For two throws of dice, the number of various pairs $\left(\xi_{1}, \xi_{2}\right)$ (where $\xi_{1}$ and $\xi_{2}$ both take one out of six values) equals to $36=6^{2}$. Generally, for $n$ throws the number of equivalent outcomes is $6^{n}$. Applying formula (1.1.1) for this number, we obtain entropy $f\left(6^{n}\right)$. According to the additivity principle, we find that
$$f\left(6^{n}\right)=n f(6)$$

## 数学代写|信息论代写information theory代考|Entropy and its properties in the case of non-equiprobable outcomes

1. Suppose now the probabilities of different outcomes are unequal. If, as earlier, the number of outcomes equals to $M$, then we can consider a random variable $\xi$, which takes one of $M$ values. Considering an index of the corresponding outcome as $\xi$, we obtain that those values are nothing else but $1, \ldots, M$. Probabilities $P(\xi)$ of those values are non-negative and satisfy the normalization constraint: $\sum_{\xi} P(\xi)=1$.

If we formally apply equality (1.1.8) to this case, then each $\xi$ should have its own entropy
$$H(\xi)=-\ln P(\xi) .$$
Thus, we attribute a certain value of entropy to each realization of the variable $\xi$. Since $\xi$ is a random variable, we can also regard this entropy as a random variable.
As in Section 1.1, the a posteriori entropy, which remains after the realization of $\xi$ becomes known, is equal to zero. That is why the information we obtain once the realization is known is numerically equal to the initial entropy
$$I(\xi)=H(\xi)=-\ln P(\xi)$$
Similar to entropy $H(\xi)$, information $I$ depends on the actual realization (on the value of $\xi$ ), i.e., it is a random variable. One can see from the latter formula that information and entropy are both large when a posteriori probability of the given realization is small and vice versa. This observation is quite consistent with intuitive ideas.

Example 1.1. Suppose we would like to know whether a certain student has passed an exam or not. Let the probabilities of these two events be
$$P(\text { pass })=7 / 8, \quad P(\text { fail })=1 / 8$$
One can see from these probabilities that the student is quite strong. If we were informed that the student had passed the exam, then we could say: ‘Your message has not given me a lot of information. I have already expected that the student passed the exam’. According to formula (1.2.2) the information of this message is quantitatively equal to
$$I(\text { pass })=\log {2}(8 / 7)=0.193 \text { bits. }$$ If we were informed that the student had failed, then we would say ‘Really?’ and would feel that we have improved our knowledge to a greater extent. The amount of information of such a message is equal to $$I(\text { fail })=\log {2}(8)=3 \text { bits. }$$

## 数学代写|信息论代写information theory代考|Definition of information and entropy in the absence of noise

$$H_{\xi}=-\sum_{\xi} P(\xi) \ln P(\xi)$$

## 数学代写|信息论代写information theory代考|Definition of entropy in the case of equiprobable outcomes

$$H=f(M),$$

$$I=H_{\mathrm{pr}}-H_{\mathrm{ps}} .$$

$$f\left(6^{n}\right)=n f(6)$$

## 数学代写|信息论代写information theory代考|Entropy and its properties in the case of non-equiprobable outcomes

1. 假设现在不同结果的概率不相等。如果如前所述，结果的数量等于 $M$ ，那么我们可以考虑一个随机变量 $\xi$ ，它需要 一个 $M$ 价值观。将相应结果的索引视为 $\xi$ ，我们得到这些值只不过是 $1, \ldots, M$. 概率 $P(\xi)$ 这些值中的一个是非负 的并且满足规范化约束: $\sum_{\xi} P(\xi)=1$.
如果我们正式将等式 (1.1.8) 应用于这种情况，那么每个 $\xi$ 应该有自己的熵
$$H(\xi)=-\ln P(\xi) .$$
因此，我们将某个熵值赋予变量的每个实现 $\xi$. 自从 $\xi$ 是一个随机变量，我们也可以把这个熵看作一个随机变量。 与第 $1.1$ 节一样，后验樀在实现 $\xi$ 变得已知，等于零。这就是为什么我们在实现已知后获得的信息在数值上等于初始樀
$$I(\xi)=H(\xi)=-\ln P(\xi)$$
类似于樀 $H(\xi)$ ，信自 $I$ 取决于实际实现（在价值 $\xi$ ，即它是一个随机变量。从后一个公式可以看出，当给定实现的后验 概率很小时，信息和樀都很大，反之亦然。这种观眎与直觉的想法是相当一致的。
例 1.1。假设我们想知道某个学生是否通过了考试。让这两个事件的概率为
$$P(\text { pass })=7 / 8, \quad P(\text { fail })=1 / 8$$
从这些概率可以看出，学生的实力相当强。如果我们被告知学生通过了考试，那么我们可以说：’你的消自没有给我很多 信息。我已经预料到学生会通过考试。”根据公式 (1.2.2)，这条消息的信息量等于
$$I(\text { pass })=\log 2(8 / 7)=0.193 \text { bits. }$$
如果殘们被告知学生失败了，那么我们会说“真的吗? “并且会觉得我们在更大程度上提高了我们的知识。这样一条消息 的信息量等于
$$I(\text { fail })=\log 2(8)=3 \text { bits. }$$

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