### 数学代写|信息论代写information theory代考|ELEN90030

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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|Conditional entropy. Hierarchical additivity

Let us generalize formulae (1.2.1), (1.2.3) to the case of conditional probabilities. Let $\xi_{1}, \ldots, \xi_{n}$ be random variables described by the joint distribution $P\left(\xi_{1}, \ldots, \xi_{n}\right)$. The conditional probabilities
$$P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)=\frac{P\left(\xi_{1}, \ldots, \xi_{n}\right)}{P\left(\xi_{1}, \ldots, \xi_{k-1}\right)} \quad(k \leqslant n)$$
are associated with the random conditional entropy
$$H\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)=-\ln P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)$$
Let us introduce a special notation for the result of averaging (1.3.1) over $\xi_{k}, \ldots, \xi_{n}$ :
\begin{aligned} H_{\xi_{k} \ldots \xi_{n}}\left(\mid \xi_{1}, \ldots \xi_{k-1}\right)=-\sum_{\xi_{k} \ldots \xi_{n}} P\left(\xi_{k}, \ldots\right.&\left., \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right) \times \ & \times \ln P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right) \end{aligned} and also for the result of total averaging:
\begin{aligned} H_{\xi_{k}, \ldots, \xi_{n}} \mid \xi_{1}, \ldots, \xi_{k-1} &=\mathbb{E}\left[H\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)\right] \ &=-\sum_{\xi_{1} \ldots \xi_{n}} P\left(\xi_{1} \ldots \xi_{n}\right) \ln P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right) \end{aligned}
If, in addition, we vary $k$ and $n$, then we will form a large number of different entropies, conditional and non-conditional, random and non-random. They are related by identities that will be considered below.

Before we formulate the main hierarchical equality (1.3.4), we show how to introduce a hierarchical set of random variables $\xi_{1}, \ldots, \xi_{n}$, even if there was just one random variable $\xi$ initially.

Let $\xi$ take one of $M$ values with probabilities $P(\xi)$. The choice of one realization will be made in several stages. At the first stage, we indicate which subset (from a full ensemble of non-overlapping subsets $E_{1}, \ldots, E_{m_{1}}$ ) the realization belongs to. Let $\xi_{1}$ be the index of such a subset. At the second stage, each subset is partitioned into smaller subsets $E_{\xi_{1} \xi_{2}}$. The second random variable $\xi_{2}$ points to which smaller subset the realization of the random variable belongs to. In turn, those smaller subsets are further partitioned until we obtain subsets consisting of a single element. Apparently, the number of nontrivial partitioning stages $n$ cannot exceed $M-1$. We can juxtapose a fixed partitioning scheme with a ‘decision tree’ depicted on Figure 1.1. Further considerations will be associated with a particular selected ‘tree’.

## 数学代写|信息论代写information theory代考|Asymptotic equivalence of non-equiprobable

The idea that the general case of non-equiprobable outcomes can be asymptotically reduced to the case of equiprobable outcomes is fundamental for information theory in the absence of noise. This idea belongs to Ludwig Boltzmann who derived formula (1.2.3) for entropy. Claude Shannon revived this idea and broadly used it for derivation of new results.

In considering this question here, we shall not try to reach generality, since these results form a particular case of more general results of Section 1.5. Consider the set of independent realizations $\eta=\left(\xi_{1}, \ldots, \xi_{n}\right)$ of a random variable $\xi=\xi_{j}$, which assumes one of two values 1 or 0 with probabilities $P[\xi=1]=p<1 / 2 ; P[\xi=$ $0]=1-p=q$. Evidently, the number of such different combinations (realizations) is equal to $2^{n}$. Let realization $\eta_{n_{1}}$ contain $n_{1}$ ones and $n-n_{1}=n_{0}$ zeros. Then its probability is given by
$$P\left(\eta_{n_{1}}\right)=p^{n_{1}} q^{n-n_{1}}$$
Of course, these probabilities are different for different $n_{1}$. The ratio $P\left(\eta_{0}\right) / P\left(\eta_{n}\right)=$ $(q / p)^{n}$ of the largest probability to the smallest one is big and increases fast with a growth of $n$. What equiprobability can we talk about then? The thing is that due to the Law of Large Numbers the number of ones $n_{1}=\xi_{1}+\cdots+\xi_{n}$ has a tendency to take values, which are close to its mean
$$\mathbb{E}\left[n_{1}\right]=\sum_{j=1}^{n} \mathbb{E}\left[\xi_{j}\right]=n \mathbb{E}\left[\xi_{j}\right]=n p$$

## 数学代写|信息论代写information theory代考|Asymptotic equiprobability and entropic stability

1. The ideas of preceding section concerning asymptotic equivalence of nonequiprobable and equiprobable outcomes can be extended to essentially more general cases of random sequences and processes. It is not necessary for random variables $\xi_{j}$ forming the sequence $\eta^{n}=\left(\xi_{1}, \ldots, \xi_{n}\right)$ to take only one of two values and to have the same distribution law $P\left(\xi_{j}\right)$. There is also no need for $\xi_{j}$ to be statistically independent and even for $\eta^{n}$ to be the sequence $\left(\xi_{1}, \ldots, \xi_{n}\right)$. So what is really necessary, the asymptotic equivalence?

In order to state the property of asymptotic equivalence of non-equiprobable and equiprobable outcomes in general terms we should use the notion of entropic stability of family of random variables.

A family of random variables $\left{\eta^{n}\right}$ is entropically stable if the ratio $H\left(\eta^{n}\right) / H_{\eta^{n}}$ converges in probability to one as $n \rightarrow \infty$. This means that whatever $\varepsilon>0, \eta>0$ are, there exists $N(\varepsilon, \eta)$ such that the inequality
$$P\left{\left|H\left(\eta^{n}\right) / H_{\eta^{n}}-1\right| \geqslant \varepsilon\right}<\eta$$
is satisfied for every $n \geqslant N(\varepsilon, \eta)$.
The above definition implies that $0<H_{\eta^{n}}<\infty$ and $H_{\eta^{n}}$ does not decrease with
$n$. Usually $H_{\eta^{n}} \rightarrow \infty$.
Asymptotic equiprobability can be expressed in terms of entropic stability in the form of the following general theorem.

Theorem 1.9. If a family of random variables $\left{\eta^{n}\right}$ is entropically stable, then the set of realizations of each random variable can be partitioned into two subsets $A_{n}$ and $B_{n}$ in such a way that

1. The total probability of realizations from subset $A_{n}$ vanishes:
$$P\left(A_{n}\right) \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty$$

## 数学代写|信息论代写information theory代考|Conditional entropy. Hierarchical additivity

$$P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)=\frac{P\left(\xi_{1}, \ldots, \xi_{n}\right)}{P\left(\xi_{1}, \ldots, \xi_{k-1}\right)} \quad(k \leqslant n)$$

$$H\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)=-\ln P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)$$

$$H_{\xi_{k}, \ldots \xi_{n}}\left(\mid \xi_{1}, \ldots \xi_{k-1}\right)=-\sum_{\xi_{k} \ldots \xi_{n}} P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right) \times \quad \times \ln P\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)$$

$$H_{\xi_{k}, \ldots, \xi_{n}} \mid \xi_{1}, \ldots, \xi_{k-1}=\mathbb{E}\left[H\left(\xi_{k}, \ldots, \xi_{n} \mid \xi_{1}, \ldots, \xi_{k-1}\right)\right]=-\sum_{\xi_{1} \ldots \xi_{n}} P\left(\xi_{1} \ldots \xi_{n}\right) \ln P\left(\xi_{k}, \ldots, \xi_{n}\right.$$

## 数学代写|信息论代写information theory代考|Asymptotic equivalence of non-equiprobable

$P[\xi=1]=p<1 / 2 ; P[\xi=0]=1-p=q$. 显然，这种不同组合（实现）的数量等于 $2^{n}$. 让实现 $\eta_{n_{1}}$ 包含 $n_{1}$ 一个和 $n-n_{1}=n_{0}$ 零。那么它的概率由下式给出
$$P\left(\eta_{n_{1}}\right)=p^{n_{1}} q^{n-n_{1}}$$

$$\mathbb{E}\left[n_{1}\right]=\sum_{j=1}^{n} \mathbb{E}\left[\xi_{j}\right]=n \mathbb{E}\left[\xi_{j}\right]=n p$$

## 数学代写|信息论代写information theory代考|Asymptotic equiprobability and entropic stability

1. 上一节关于非等概率和等概率结果的渐近等价的想法可以扩展到更一般的随机序列和过程的情况。随机变量 不是必需的 $\xi_{j}$ 形成序列 $\eta^{n}=\left(\xi_{1}, \ldots, \xi_{n}\right)$ 只取两个值之一并具有相同的分布规律 $P\left(\xi_{j}\right)$. 也没有必要 $\xi_{j}$ 在统 计上是独立的，甚至对于 $\eta^{n}$ 成为序列 $\left(\xi_{1}, \ldots, \xi_{n}\right)$. 那么什么是真正必要的，渐近等价呢?
为了概括地说明非等概率和等概率结果的渐近等价性质，我们应该使用随机变量族的嫡稳定性的概念。
随机变量族 Veft {leta^{n}\right } } \text { 如果比率是樀稳定的 } H ( \eta ^ { n } ) / H _ { \eta ^ { n } } \text { 在概率上收敛为 } 1 n \rightarrow \infty \text { . 这意味着无论 } $\varepsilon>0, \eta>0$ 是，存在 $N(\varepsilon, \eta)$ 这样不等式
P\left{\eft $\mid H \backslash l e f t(\mathrm{~ l e t a}$
满足于每一个 $n \geqslant N(\varepsilon, \eta)$.
上述定义意味着 $0<H_{\eta^{n}}<\infty$ 和 $H_{\eta^{n}}$ 不减少
$n$. 通常 $H_{\eta^{n}} \rightarrow \infty$.
渐近等概率可以用以下一般定理的形式用嫡稳定性表示。
$\mathrm{~ 定 理 ~ 1 . 9 。 如 果 一 个 随 机 变 量 族 l l e f t : n e t a ^ { n }}$ $A_{n}$ 和 $B_{n}$ 以这样的方式
2. 子集实现的总概率 $A_{n}$ 消失:
$$P\left(A_{n}\right) \rightarrow 0 \quad \text { as } \quad n \rightarrow \infty$$

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