# 数学代写|信息论代写information theory代考|EXAMPLES OF SANOV’S THEOREM

#### Doug I. Jones

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## 数学代写|信息论代写information theory代考|EXAMPLES OF SANOV’S THEOREM

Suppose that we wish to find $\operatorname{Pr}\left{\frac{1}{n} \sum_{i=1}^n g_j\left(X_i\right) \geq \alpha_j, j=1,2, \ldots, k\right}$. Then the set $E$ is defined as
$$E=\left{P: \sum_a P(a) g_j(a) \geq \alpha_j, j=1,2, \ldots, k\right} .$$
To find the closest distribution in $E$ to $Q$, we minimize $D(P | Q)$ subject to the constraints in (11.108). Using Lagrange multipliers, we construct the functional
$$J(P)=\sum_x P(x) \log \frac{P(x)}{Q(x)}+\sum_i \lambda_i \sum_x P(x) g_i(x)+v \sum_x P(x) .$$
We then differentiate and calculate the closest distribution to $Q$ to be of the form
$$P^(x)=\frac{Q(x) e^{\sum_i \lambda_i g_i(x)}}{\sum_{a \in \mathcal{X}} Q(a) e^{\sum_i \lambda_i g_i(a)}},$$ where the constants $\lambda_i$ are chosen to satisfy the constraints. Note that if $Q$ is uniform, $P^$ is the maximum entropy distribution. Verification that $P^*$ is indeed the minimum follows from the same kinds of arguments as given in Chapter 12.

## 数学代写|信息论代写information theory代考|CONDITIONAL LIMIT THEOREM

It has been shown that the probability of a set of types under a distribution $Q$ is determined essentially by the probability of the closest element of the set to $Q$; the probability is $2^{-n D^}$ to first order in the exponent, where $$D^=\min _{P \in E} D(P | Q) .$$
This follows because the probability of the set of types is the sum of the probabilities of each type, which is bounded by the largest term times the number of terms. Since the number of terms is polynomial in the length of the sequences, the sum is equal to the largest term to first order in the exponent.

We now strengthen the argument to show that not only is the probability of the set $E$ essentially the same as the probability of the closest type $P^$ but also that the total probability of other types that are far away from $P^$ is negligible. This implies that with very high probability, the type observed is close to $P^*$. We call this a conditional limit theorem.

Before we prove this result, we prove a “Pythagorean” theorem, which gives some insight into the geometry of $D(P | Q)$. Since $D(P | Q)$ is not a metric, many of the intuitive properties of distance are not valid for $D(P | Q)$. The next theorem shows a sense in which $D(P | Q)$ behaves like the square of the Euclidean metric (Figure 11.5).

Theorem 11.6.1 For a closed convex set $E \subset \mathcal{P}$ and distribution $Q \notin$ $E$, let $P^* \in E$ be the distribution that achieves the minimum distance to $Q$; that is,
$$D\left(P^* | Q\right)=\min _{P \in E} D(P | Q) .$$
Then
$$D(P | Q) \geq D\left(P | P^\right)+D\left(P^ | Q\right)$$
for all $P \in E$.

# 信息论代写

## 数学代写|信息论代写information theory代考|EXAMPLES OF SANOV’S THEOREM

$$E=\left{P: \sum_a P(a) g_j(a) \geq \alpha_j, j=1,2, \ldots, k\right} .$$

$$J(P)=\sum_x P(x) \log \frac{P(x)}{Q(x)}+\sum_i \lambda_i \sum_x P(x) g_i(x)+v \sum_x P(x) .$$

$$P^(x)=\frac{Q(x) e^{\sum_i \lambda_i g_i(x)}}{\sum_{a \in \mathcal{X}} Q(a) e^{\sum_i \lambda_i g_i(a)}},$$，其中选择常数$\lambda_i$以满足约束。注意，如果$Q$是均匀的，则$P^$是最大熵分布。证明$P^*$确实是最小值，可以从第12章给出的相同类型的论证中得到。

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## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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