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## 数学代写|信息论代写information theory代考|Asymptotic equivalence of two types of constraints

Consider asymptotic results related to the content of Chapter 3 . We will show that when computing maximum entropy (capacity of a noiseless channel), constraints imposed on mean values and constraints imposed on exact values are asymptotically equivalent to each other. These results are closely related to the theorem about stability of the canonical distribution proven in Section 4.2.

We set out the material in a more general form than in Sections $3.2$ and $3.3$ by using an auxiliary measure $v(d x)$ similarly to Section 3.6.

Let the space $X$ with measure $v(d x)$ (not normalized to unity) be given. Entropy will be defined by the formula $$H=-\int \ln \frac{P(d x)}{v(d x)} P(d x)$$
[see (1.6.13)]. Let the constraint
$$B(x) \leqslant A$$
or, more generally,
$$B(x) \in E$$
be given, where $E$ is some (measurable) set, $B(x)$ is a given function. Entropy of level $A($ or set $E$ ) is defined by maximization
$$\tilde{H}=\sup _{P \in \tilde{G}}\left[-\int P(d x) \ln \frac{P(d x)}{v(d x)}\right]$$
Here set $\widetilde{G}$ of feasible distributions $P(\cdot)$ is characterized by the fact that a probability is concentrated within subspace $\widetilde{X} \subset X$ defined by constraints (4.3.2) and (4.3.3), i.e.
$$P(\tilde{X})=1 ; \quad P[B(x) \in E]=1 .$$
Constraints (4.3.2), (4.3.3) will be relaxed if we substitute them by analogous constraints for expectations:
$$\mathbb{E}[B(x)] \leqslant A, \quad \text { where } \quad \mathbb{E}[B(x)] \in E$$

## 数学代写|信息论代写information theory代考|Some theorems about the characteristic potential

1. Characteristic potential $\mu(s)=\ln \Theta(s)$ of random variables $B_{i}(\xi)$ (or a cumulant generating function) was defined by formula (4.1.11a) earlier. If there is given the family of distributions $p(\xi \mid \alpha)=\exp \left[-\Gamma(\alpha)+\alpha B(\xi)-\varphi_{0}(\xi)\right]$, then $\mu(s)$ is expressed in terms of $\Gamma(\alpha)$ by formula (4.1.12a). If there is merely a given random variable $\xi$ with probability distribution $P(d \xi)$ instead of a family of distributions, then we can construct the following family of distributions:
$$P(d \xi \mid \alpha)=\text { conste } e^{\alpha B(\xi)} P(d \xi) .$$
Because of (4.1.11a) the normalization constant is expressed through $\mu(\alpha)$, so that
$$\begin{gathered} P(d \xi \mid \alpha)=\exp [-\mu(\alpha)+\alpha B(\xi)] P(d \xi) \ \left(\ln p(\xi)=-\varphi_{0}(\xi)\right) . \end{gathered}$$
Thus, we have built the family ${P(d \xi \mid \alpha), \alpha \in Q}$ on basis of $P(\xi)$. Besides the characteristic potential $\mu(s)$ of the initial distribution, we can find a characteristic potential of any distribution (4.4.1) from the indicated family by formula (4.1.12a), i.e. by the formula
$$\mu(s \mid \alpha)=\mu(\alpha+s)-\mu(\alpha),$$
2. At first consider an easy example of a single variable $B(\xi), r=1$ and prove a simple but useful theorem.

## 数学代写|信息论代写information theory代考|Computation of entropy for special cases

In the present chapter, we set out the methods for computation of entropy of many random variables or of a stochastic process in discrete and continuous time.

From a fundamental and practical points of view, of particular interest are the stationary stochastic processes and their information-theoretic characteristics, specifically their entropy. Such processes are relatively simple objects, particularly a discrete process, i.e. a stationary process with discrete states and running in discrete time. Therefore, this process is a very good example for demonstrating the basic points of the theory, and so we shall start from its presentation.

Our main attention will be drawn to the definition of such an important characteristic of a stationary process as the entropy rate, that is entropy per unit of time or per step. In addition, we introduce entropy $\Gamma$ at the end of an interval. This entropy together with the entropy rate $H_{1}$ defines the entropy of a long interval of length $T$ by the approximate formula
$$H_{T} \approx H_{1} T+2 \Gamma,$$
which is the more precise, the greater is $T$. Both constants $H_{1}$ and $\Gamma$ are calculated for a discrete Markov process.

The generalized definition of entropy, given in Section 1.6, allows for the application of this notion to continuous random variables as well as to the case when the set of these random variables is continuum, i.e. to a stochastic process with a continuous parameter (time).

In what follows, we show that many results related to a discrete process can be extended both to the case of continuous sample space and to continuous time. For instance, we can introduce the entropy rate (not per one step but per a unit of time) and entropy of an end of an interval for continuous-time stationary processes. The entropy of a stochastic process on an interval is represented approximately in the form of two terms by analogy with the aforementioned formula.

For non-stationary continuous-time processes, instead of constant entropy rate, one should consider entropy density, which, generally speaking, is not constant in time.

## 数学代写|信息论代写information theory代考|Asymptotic equivalence of two types of constraints

$$H=-\int \ln \frac{P(d x)}{v(d x)} P(d x)$$
[见 (1.6.13)]。让约束
$$B(x) \leqslant A$$

$$B(x) \in E$$

$$\tilde{H}=\sup _{P \in \bar{G}}\left[-\int P(d x) \ln \frac{P(d x)}{v(d x)}\right]$$

$$P(\tilde{X})=1 ; \quad P[B(x) \in E]=1 .$$

$$\mathbb{E}[B(x)] \leqslant A, \quad \text { where } \quad \mathbb{E}[B(x)] \in E$$

## 数学代写|信息论代写information theory代考|Some theorems about the characteristic potential

1. 特征潜力 $\mu(s)=\ln \Theta(s)$ 随机变量 $B_{i}(\xi)$ (或累积量生成函数) 由公式 (4.1.11a) 定义。如果给定分布族 $p(\xi \mid \alpha)=\exp \left[-\Gamma(\alpha)+\alpha B(\xi)-\varphi_{0}(\xi)\right]$ ，然后 $\mu(s)$ 表示为 $\Gamma(\alpha)$ 由公式 (4.1.12a) 。如果只有一个给定的随机变量 $\xi$ 有概率分布 $P(d \xi)$ 而不是一个分布族，那么我们可以构造以下分布族：
2. $$3. P(d \xi \mid \alpha)=\text { conste } e^{\alpha B(\xi)} P(d \xi) . 4.$$
5. 由于 (4.1.11a)，归一化常数表示为 $\mu(\alpha)$ ，以便
6. $$7. P(d \xi \mid \alpha)=\exp [-\mu(\alpha)+\alpha B(\xi)] P(d \xi)\left(\ln p(\xi)=-\varphi_{0}(\xi)\right) . 8.$$
9. 就这样，我们建立了家庭 $P(d \xi \mid \alpha), \alpha \in Q$ 基于 $P(\xi)$. 除了特征潜力 $\mu(s)$ 对于初始分布，我们可以通过公式 (4.1.12a) 从指定的族 中找到任何分布 (4.4.1) 的特征势，即通过公式
10. $$11. \mu(s \mid \alpha)=\mu(\alpha+s)-\mu(\alpha), 12.$$
13. 首先考虑一个简单的单变量示例 $B(\xi), r=1$ 并证明一个简单但有用的定理。

## 数学代写|信息论代写information theory代考|Computation of entropy for special cases

$$H_{T} \approx H_{1} T+2 \Gamma$$

$1.6$ 节给出的樀的广义定义允许将此概念应用于连续随机变量以及这些随机变量的集合是连续的情况，即具有连续参数（时间) 的随机过 程.

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