Blog | XHT | Haotian Xue - AI Research & Robotics

Day 1｜信息论：熵、KL 散度与互信息

Tue, 11 Aug 2026 00:00:00 +0000

1.1 熵

一种混乱程度具有较高的熵，认为具有较高的信息含量

1.2 离散变量

1.2.1 离散变量的熵

$K$ 个状态上分布为 $p$ 的离散随机变量 $X$ 的熵被定义为

$$ \mathbb{H}(x)\triangleq-\sum_{k=1}^{K}p(X=k)\log_2p(X=k)=-\mathbb{E}_X\left[\log p(X)\right] $$

1.2.2 重要的性质

具有最大熵的离散分布是均匀分布
具有最小熵的分布是任何将其所有概率质量放在一个状态上的$\delta$函数

证明在介绍完KL散度后证明

1.3 交叉熵

分布 $p$ 和 $q$ 的交叉熵被定义为

$$ \mathbb{H}(p,q)\triangleq-\sum_{k=1}^{K}p_k\log q_k $$

特别的，对于固定分布 $p$，当 $q=p$ 时，交叉熵取得最小值

1.4 联合熵

两个随机变量 $X$ 和 $Y$ 的联合熵可以被定义为

$$ \mathbb{H}(X,Y)=-\sum_{x,y}p(x,y)\log_2p(x,y) $$

1.5 条件熵

当给定 $X$ 时， $Y$ 的条件熵是在观察到 $X$ 后，对 $Y$ 的不确定性在 $X$ 的所有可能值熵取平均值

$$ \begin{align*} \mathbb{H}(Y|X)&\triangleq\mathbb{E}_{p(x)}\left[\mathbb{H}(p(Y|X))\right]\\ &=\sum_{x}p(x)\mathbb{H}(p(Y|X=x))=-\sum_{x}p(x)\sum_{y}p(y|x)\log p(y|x)\\ &=-\sum_{x,y}p(x,y)\log p(y|x)=-\sum_{x,y}p(x,y)\log \frac{p(x,y)}{p(x)}\\ &=-\sum_{x,y}p(x,y)\log p(x,y)-\sum_{x}p(x)\log \frac{1}{p(x)}\\ &=\mathbb{H}(X,Y)-\mathbb{H}(X) \end{align*} $$

我们可以令 $X=X_1$ 、$Y=X_2$，则有$\mathbb{H}(X_1,X_2)=\mathbb{H}(X_1)+\mathbb{H}(X_2|X_1)$，进而有熵的链式法则

$$ \mathbb{H}(X_1,X_2,...,X_n)=\sum_{i=1}^{n}\mathbb{H}(X_i|X_1,...,X_{i-1}) $$

1.6 困惑度

离散概率分布 $p$ 的困惑度定义为

$$ \text{perplexity}\triangleq2^{\mathbb{H}(p)} $$

1.7 连续变量的微分熵

若 $X$ 是一个具有概率密度函数 $p(X)$ 的连续随机变量，我们将微分熵定义为

$$ h(x)\triangleq-\int_X p(x)\log p(x) dx $$

1.8 相对熵

对于离散分布，我们使用 $\mathbb{KL}$ 散度进行定义

$$ \mathbb{KL}(p\|q)\triangleq\sum_{k=1}^{K}p_k \log \frac{p_k}{q_k} $$

对于连续分布

$$ \mathbb{KL}(p\|q)\triangleq\int p(x) \log \frac{p(x)}{q(x)}dx $$

进一步可以解释为：

$$ \mathbb{KL}(p\|q)=\underbrace{\sum_{k=1}^{K}p_k \log p_k}_{-\mathbb{H}(p)}-\underbrace{\sum_{k=1}^{K}p_k \log q_k}_{-\mathbb{H}(p,q)} $$

$$ \mathbb{KL}(p\|q)=-\mathbb{H}(p)+\mathbb{H}(p,q) $$

$\mathbb{KL}$ 散度用于测量两个分布相似程度的方法

1.8.1 重要证明

证明1：$\mathbb{KL}$ 散度非负
$$ \begin{align*} \mathbb{KL}(p\|q)&=\sum_{i}p_i \log \frac{p_i}{q_i}\\ &=-\sum_{i}p_i \log \frac{q_i}{p_i} \end{align*} $$
由于 $\log x\le x−1$ ，我们可以得到 $\log\frac{q_i}{p_i}\le \frac{q_i}{p_i}−1$，然后化简有 $p_i\log\frac{q_i}{p_i}\le q_i−p_i$
$$ \begin{align*} -\mathbb{KL}(p\|q)&\le \sum_{i}(q_i-p_i)\\ &\le 0 \end{align*} $$
由于概率和为1，则有 $-\mathbb{KL}(p\|q) \le 0$，所以 $\mathbb{KL}(p\|q) \ge 0$。
证明2：具有最大熵的离散分布是均匀分布首先，令 $u_i=\frac{1}{K}$，于是均匀分布为 $u=\big(\frac{1}{K},...,\frac{1}{K}\big)$。考虑概率分布 $p$ 与均匀分布 $u$ 之间的 KL 散度：
$$ \begin{align*} \mathbb{KL}(p\|u)&=\sum_{i=1}^{K}p_i\log\frac{p_i}{u_i}\\ &=\sum_{i=1}^{K}p_i\log\frac{p_i}{1/K}\\ &=\sum_{i=1}^{K}p_i\big(\log p_i+\log K\big)\\ &=\sum_{i=1}^{K}p_i\log p_i+\log K\sum_{i=1}^{K}p_i \end{align*} $$
由于$\sum_{i=1}^{K}p_i=1$,
$$ \begin{align*} \mathbb{KL}(p\|u)&=\sum_{i=1}^{K}p_i\log p_i+\log K\\ &=\log K-\mathbb{H}(p) \end{align*} $$
由于 $\mathbb{KL}$ 散度非负，$\log K\ge\mathbb{H}(p)$，当且仅当两个分布相等时，取等号使得熵最大，所以有限离散状态空间上，均匀分布拥有最大熵 $Q.E.D.$
证明3：具有最小熵的分布是任何将其所有概率质量放在一个状态上的 $\delta$ 函数对于离散随机变量，$\delta$ 分布表示所有概率质量都集中在某一个状态 $x_0$ 上：
$$ p(x)=\left\{\begin{matrix} 1,&x=x_0 \\ 0,&x\ne x_0 \end{matrix}\right. $$
离散随机变量的熵为
$$ H(p)=-\sum_x p(x)\log p(x) $$
由于对任意概率 $0\le p(x)\le1$，都有
$$ \log p(x)\le0 $$
因此
$$ -p(x)\log p(x)\ge0 $$
所以
$$ H(p)\ge0 $$
也就是说，离散熵的最小可能值为 $0$。要使
$$ H(p)=0 $$
由于熵中的每一项都非负，因此必须对所有 $x$ 都有
$$ -p(x)\log p(x)=0 $$
而当 $0 $$ -p(x)\log p(x)>0 $$
所以只有
$$ p(x)=0 \quad\text{或}\quad p(x)=1 $$
才能使该项为零。再结合概率归一化条件
$$ \sum_x p(x)=1 $$
只能有一个状态的概率为 $1$，其余状态的概率全部为 $0$。因此，
$$ \boxed{H_{\min}=0} $$
且最小值恰好由任意 $\delta$ 分布取得。 $Q.E.D.$

1.9 互信息

衡量两个随机变量的相关性随机变量 $X$ 和 $Y$ 之间的互信息定义为：

$$ \mathbb{I}(X;Y)\triangleq\mathbb{KL}(p(x,y)\|p(x)p(y))=\sum_{y\in Y}\sum_{x\in X}p(x,y)\log \frac{p(x,y)}{p(x)p(y)} \ge 0 $$

当且仅当 $p(x,y)=p(x)p(y)$ 时，达到边界0 进一步可以使用联合熵和条件熵来表示

$$ \mathbb{I}(X;Y)=\mathbb{H}(X)-\mathbb{H}(X|Y)=\mathbb{H}(Y)-\mathbb{H}(Y|X) $$

进一步化简合并可以得到

$$ \mathbb{I}(X;Y)=\mathbb{H}(X) + \mathbb{H}(Y)-\mathbb{H}(X,Y) $$

然后从韦恩图上我们可以更加明显的理解这几个概念

1.9.1 条件互信息

当然我们还可以定义条件互信息

$$ \begin{align*} \mathbb{I}(X; Y|Z) &\triangleq \mathbb{E}_{p(z)}\left[\mathbb{I}(X; Y|Z=z)\right]\\ &= \mathbb{E}_{p(x, y, z)} \left[ \log \frac{p(x, y|z)}{p(x|z)p(y|z)} \right]\\ &= \mathbb{H}(X|Z) + \mathbb{H}(Y|Z) - \mathbb{H}(X, Y|Z)\\ &= \mathbb{H}(X|Z) - \mathbb{H}(X|Y, Z) = \mathbb{H}(Y|Z) - \mathbb{H}(Y|X, Z)\\ &= \mathbb{H}(X, Z) + \mathbb{H}(Y, Z) - \mathbb{H}(Z) - \mathbb{H}(X, Y, Z)\\ &= \mathbb{I}(Y; X, Z) - \mathbb{I}(Y; Z) \end{align*} $$

上式也可以写为 $\mathbb{I}(Z, Y; X) = \mathbb{I}(Z; X) + \mathbb{I}(Y; X|Z)$，进而有互信息的链式法则

$$ \mathbb{I}(Z_1, \cdots, Z_N; X) = \sum_{n=1}^N \mathbb{I}(Z_n; X | Z_1, \cdots, Z_{n-1}) $$

1.9.2 归一化互信息

当分母非零时，我们可以定义一个介于 $0$ 和 $1$ 之间的归一化相关性度量

$$ \text{NMI}(X,Y)=\frac{\mathbb{I}(X;Y)}{\text{min}(\mathbb{H}(X),\mathbb{H}(Y))} $$

1.9.3 最大信息系数

是一种用于衡量两个变量之间相关性的统计量，可以用于发现线性关系以及多种非线性关系。

MIC 的基本思想是：将二维样本空间划分成不同大小的网格，在各种网格划分中寻找能够产生最大互信息的划分，并对互信息进行归一化。

对于数据集 $D$，给定一个 $x\times y$ 的网格，定义

$$ M(D)_{x,y}=\frac{\max_{G\in\mathcal{G}_{x,y}} I(D|G)}{\log \min{\{x,y\}}} $$

其中，$\mathcal{G}_{x,y}$ 表示所有可能的 $x\times y$ 网格划分，$I(D|G)$ 表示数据 $D$ 在网格 $G$ 下离散化后得到的互信息。

分母

$$ \log \min{\{x,y\}} $$

用于对互信息进行归一化，因为对于一个 $x\times y$ 的网格，

$$ I(X;Y) \le \min{H(X),H(Y)}\le \log \min\{{x,y\}} $$

因此 $M(D)_{x,y}$ 可以理解为：当前网格能够捕获的信息量占该网格理论最大信息量的比例。 最后，在所有满足复杂度约束的网格中取最大值：

$$ \operatorname{MIC}(D)=\max_{xy 其中 $n$ 为样本数量，$B(n)$ 用于限制网格复杂度，防止网格划分过细而产生过拟合。因此，MIC 可以简单理解为不同网格划分下的最大归一化互信息

MIC 的取值通常位于 $0$ 到 $1$ 之间。MIC 越接近 $1$，说明两个变量之间存在越强的依赖关系；越接近 $0$，说明能够检测到的依赖关系越弱。

1.9.4 数据处理不等式

假设有一个位置变量 $X$，我们观察到该未知变量的一个噪声函数，称之为 $Y$ 。如果我们以某种方式处理有噪声的观测结果，以创建一个新的变量 $Z$ ，那么显然，我们无法增加关于未知变量 $X$ 的信息量，这个就被成为数据处理不等式形式化表达：假设 $X\rightarrow Y\rightarrow Z$ 形成一个马尔可夫链，那么在给定 $Y$ 的条件下 $X$ 、$Z$ 相互独立，则$\mathbb{I}(X;Y)\ge\mathbb{I}(X;Z)$ 。证明：由于互信息的链式法则，我们可以使用两种不同的方式扩展互信息

$$ \mathbb{I}(X;Y,Z)=\mathbb{I}(X;Z)+\mathbb{I}(X;Y|Z)=\mathbb{I}(X;Y)+\mathbb{I}(X;Z|Y) $$

由于在给定 $Y$ 的条件下 $X$ 、$Z$ 相互独立，因此 $\mathbb{I}(X;Z|Y)=0$，于是

$$ \mathbb{I}(X;Z)+\mathbb{I}(X;Y|Z)=\mathbb{I}(X;Y) $$

由于 $\mathbb{I}(X;Y|Z)\ge0$ ,因此 $\mathbb{I}(X;Y)\ge\mathbb{I}(X;Z)$，同理可以证明 $\mathbb{I}(Y;Z)\ge\mathbb{I}(X;Z)$。

1.9.5 充分统计量

从数据处理不等式中我们可以得到重要结论，假设 $\theta\rightarrow\mathcal{D}\rightarrow s(\mathcal{D})$，于是

$$ \mathbb{I}(\theta;s(\mathcal{D}))\le\mathbb{I}(\theta;\mathcal{D}) $$

若等式成立，那么我们称 $s(\mathcal{D})$ 是数据 $\mathcal{D}$ 用于推断 $\theta$ 的充分统计量，一个例子，$s(\mathcal{D}) =\mathcal{D}$，就是数据本身。进而我们可以定义一个最小充分统计量：如果对于所有充分统计量 $s'(\mathcal{D})$ ，存在某个函数 $f$ ，使得 $s(\mathcal{D})=f(s'(\mathcal{D}))$ ，那么我们称 s 是 D 的最小充分统计量

Welcome to My AI Research Blog: Multi-Agent Systems, Robotics and World Models

Sun, 02 Aug 2026 00:00:00 +0000

Welcome to my personal website and blog.

My name is Haotian Xue. I am currently a graduate student in Software Engineering at Sichuan University, where I am exploring topics at the intersection of artificial intelligence, autonomous systems, robotics, and software engineering.

I created this website as a place to organize my work, document what I learn, and share ideas that may be useful to other students, researchers, and developers.

Why I Started This Blog

Research and software development involve much more than presenting a final result. Behind every paper, experiment, or open-source project, there are usually many intermediate steps:

understanding a new problem;
reading papers and technical documentation;
designing experiments;
debugging code and environments;
comparing alternative methods;
recording failed attempts;
and gradually refining an idea.

Many of these details are valuable, but they are often scattered across notebooks, repositories, documents, and temporary files. This blog gives me a more structured place to preserve them.

Writing also helps me examine whether I truly understand a topic. When an idea cannot be explained clearly, it often means that some part of the reasoning is still incomplete. For this reason, I hope this website will serve not only as a public portfolio, but also as a long-term record of my learning and research process.

What I Am Interested In

My current interests include several closely related areas.

Multi-Agent Reinforcement Learning

Multi-agent reinforcement learning studies how multiple decision-making agents learn to cooperate, compete, communicate, and coordinate in shared environments.

I am especially interested in problems involving structured environments, decentralized decision-making, safety constraints, heterogeneous agents, and generalization to unseen scenarios.

These problems appear in many real-world systems, including intelligent transportation, robotics, autonomous vehicles, and large-scale coordination.

World Models

World models aim to learn compact representations of an environment and predict how that environment may evolve.

A useful world model can help an intelligent agent reason about future states before taking an action. This creates opportunities for planning, imagination-based learning, sample-efficient reinforcement learning, and long-horizon decision-making.

I am interested in how world models can be extended from single-agent settings to multi-agent and embodied systems.

Robotics and Autonomous Systems

Robotics connects perception, planning, control, and interaction with the physical world.

I am particularly interested in autonomous systems that must operate under uncertainty, respond to changing environments, and coordinate with other agents. This includes aerial robots, intelligent vehicles, and other embodied platforms.

Intelligent Transportation

Transportation systems naturally involve many interacting agents, shared resources, safety requirements, and complex operational constraints.

I am interested in how learning-based methods can be combined with graph representations, domain knowledge, planning algorithms, and safety rules to improve coordination and decision-making in these systems.

Open-Source Software

Open-source development is also an important part of my learning process.

Implementing ideas, organizing code, reproducing experiments, and documenting software are essential skills for turning research concepts into usable systems. I plan to share development records, environment setup notes, debugging experiences, and lessons learned from my projects.

The content of this blog will mainly include:

research notes on artificial intelligence and reinforcement learning;
paper reading notes and method explanations;
technical tutorials and environment setup guides;
experiment design and implementation records;
project development summaries;
debugging notes and solutions to practical problems;
reflections on research, learning, and open-source development.

Some posts will focus on theoretical ideas, while others will be more practical and code-oriented. I hope to keep the explanations clear and reproducible whenever possible.

My Background

Before beginning my graduate study at Sichuan University, I completed a bachelor’s degree in Computer Science and Technology at the Civil Aviation University of China.

My previous studies and projects involved software engineering, systems programming, computer networks, and open-source development. These experiences gradually led me toward artificial intelligence, autonomous systems, and multi-agent decision-making.

I am still learning, and this website will evolve together with my research interests and technical experience.

Looking Ahead

This is the first post on the website, but it is not intended to remain only a simple welcome page.

Over time, I hope the blog will become a searchable collection of useful research notes, technical explanations, project records, and lessons from both successful and unsuccessful experiments.

You can also find my open-source work on and some of my earlier technical writing on my .

Thank you for visiting.