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蒙特卡罗方法与人工智能

作译者：魏平

出版时间：2024-01 千字数：550 版次：01-01 页数：344

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目 录
第1 章 蒙特卡罗方法简介··············································································.1
1.1 引言·······························································································.1
1.2 动机和目标······················································································.1
1.3 蒙特卡罗计算中的任务·······································································.2
1.3.1 任务1：采样和模拟········································································.3
1.3.2 任务2：通过蒙特卡罗模拟估算未知量···················································.5
1.3.3 任务3：优化和贝叶斯推理································································.7
1.3.4 任务4：学习和模型估计···································································.8
1.3.5 任务5：可视化能级图·····································································.9
本章参考文献··························································································13
第2 章 序贯蒙特卡罗方法··············································································14
2.1 引言·······························································································14
2.2 一维密度采样···················································································14
2.3 重要性采样和加权样本·······································································15
2.4 序贯重要性采样（SIS） ······································································18
2.4.1 应用：表达聚合物生长的自避游走························································18
2.4.2 应用：目标跟踪的非线性/粒子滤波·······················································20
2.4.3 SMC 方法框架总结·········································································23
2.5 应用：利用SMC 方法进行光线追踪·······················································24
2.6 在重要性采样中保持样本多样性···························································25
2.6.1 基本方法····················································································25
2.6.2 Parzen 窗讨论··············································································28
2.7 蒙特卡罗树搜索················································································29
2.7.1 纯蒙特卡罗树搜索··········································································30
2.7.2 AlphaGo ·····················································································32
2.8 本章练习·························································································33
本章参考文献··························································································35
第3 章 马尔可夫链蒙特卡罗方法基础·······························································36
3.1 引言·······························································································36
蒙特卡罗方法与人工智能
·X ·
3.2 马尔可夫链基础················································································37
3.3 转移矩阵的拓扑：连通与周期······························································38
3.4 Perron-Frobenius 定理··········································································41
3.5 收敛性度量······················································································42
3.6 连续或异构状态空间中的马尔可夫链·····················································44
3.7 各态遍历性定理················································································45
3.8 通过模拟退火进行MCMC 优化·····························································46
3.9 本章练习·························································································49
本章参考文献··························································································51
第4 章 Metropolis 算法及其变体······································································52
4.1 引言·······························································································52
4.2 Metropolis-Hastings 算法······································································52
4.2.1 原始Metropolis-Hastings 算法······························································53
4.2.2 Metropolis-Hastings 算法的另一形式·······················································54
4.2.3 其他接受概率设计··········································································55
4.2.4 Metropolis 算法设计中的关键问题·························································55
4.3 独立Metropolis 采样···········································································55
4.3.1 IMS 的特征结构············································································56
4.3.2 有限空间的一般首中时·····································································57
4.3.3 IMS 击中时分析············································································57
4.4 可逆跳跃和跨维MCMC ······································································59
4.4.1 可逆跳跃····················································································59
4.4.2 简单例子：一维图像分割··································································60
4.5 应用：计算人数················································································63
4.5.1 标值点过程模型············································································64
4.5.2 MCMC 推理·················································································64
4.5.3 结果·························································································65
4.6 应用：家具布置················································································65
4.7 应用：场景合成················································································67
4.8 本章练习·························································································71
本章参考文献··························································································72
第5 章 吉布斯采样器及其变体········································································73
5.1 引言·······························································································73
5.2 吉布斯采样器···················································································74
目 录
·XI·
5.2.1 吉布斯采样器介绍··········································································74
5.2.2 吉布斯采样器的一个主要问题·····························································75
5.3 吉布斯采样器扩展·············································································76
5.3.1 击中逃跑····················································································77
5.3.2 广义吉布斯采样器··········································································77
5.3.3 广义击中逃跑···············································································77
5.3.4 利用辅助变量采样··········································································78
5.3.5 模拟退火····················································································78
5.3.6 切片采样····················································································79
5.3.7 数据增强····················································································80
5.3.8 Metropolized 吉布斯采样器·································································80
5.4 数据关联和数据增强··········································································82
5.5 Julesz 系综和MCMC 纹理采样······························································83
5.5.1 Julesz 系综：纹理的数学定义······························································84
5.5.2 吉布斯系综和系综等价性··································································85
5.5.3 Julesz 系综采样·············································································86
5.5.4 实验：对Julesz 系综进行采样·····························································87
5.6 本章练习·························································································89
本章参考文献··························································································90
第6 章 聚类采样方法····················································································91
6.1 引言·······························································································91
6.2 Potts 模型和SW 算法·········································································92
6.3 SW 算法详解····················································································94
6.3.1 解释1：Metropolis-Hastings 观点··························································94
6.3.2 解释2：数据增强··········································································97
6.4 SW 算法的相关理论结果··································································.100
6.5 任意概率的SW 切分算法·································································.102
6.5.1 步骤一：数据驱动的聚类·······························································.102
6.5.2 步骤二：颜色翻转·······································································.103
6.5.3 步骤三：接受翻转·······································································.104
6.5.4 复杂性分析···············································································.105
6.6 聚类采样方法的变体·······································································.106
6.6.1 聚类吉布斯采样：“击中逃跑”观点·····················································.106
6.6.2 多重翻转方案············································································.107
6.7 应用：图像分割·············································································.107
蒙特卡罗方法与人工智能
·X II·
6.8 多重网格和多级SW 切分算法···························································.110
6.8.1 多重网格SW 切分算法··································································.111
6.8.2 多级SW 切分算法·······································································.113
6.9 子空间聚类···················································································.114
6.9.1 通过SW 切分算法进行子空间聚类·····················································.115
6.9.2 应用：稀疏运动分割····································································.117
6.10 C 4：聚类合作竞争约束··································································.121
6.10.1 C 4 算法综述············································································.123
6.10.2 图形、耦合和聚类······································································.124
6.10.3 平面图上的C 4 算法····································································.128
6.10.4 在平面图上的实验······································································.131
6.10.5 棋盘Ising 模型·········································································.132
6.10.6 分层图上的C 4··········································································.136
6.10.7 C 4 分层实验············································································.138
6.11 本章练习·····················································································.139
本章参考文献·······················································································.140
第7 章 MCMC 的收敛性分析·······································································.144
7.1 引言····························································································.144
7.2 关键收敛问题················································································.144
7.3 实用的监测方法·············································································.145
7.4 洗牌的耦合方法·············································································.146
7.4.1 置顶洗牌·················································································.147
7.4.2 Riffle 洗牌················································································.147
7.5 几何界限、瓶颈和连通率·································································.149
7.5.1 几何收敛·················································································.149
7.5.2 交易图（转换图）·······································································.150
7.5.3 瓶颈······················································································.150
7.5.4 连通率····················································································.151
7.6 Peskun 有序和遍历性定理·································································.152
7.7 路径耦合和精确采样·······································································.153
7.7.1 从过去耦合···············································································.154
7.7.2 应用：对Ising 模型进行采样···························································.155
7.8 本章练习······················································································.157
本章参考文献·······················································································.159
目 录
·XIII·
第8 章 数据驱动的马尔可夫链蒙特卡罗方法···················································.160
8.1 引言····························································································.160
8.2 图像分割和DDMCMC 方法概述························································.160
8.3 DDMCMC 方法解释········································································.161
8.3.1 MCMC 方法设计的基本问题····························································.163
8.3.2 计算原子空间中的提议概率：原子粒子················································.164
8.3.3 计算对象空间中的提议概率：对象粒子················································.166
8.3.4 计算多个不同的解：场景粒子··························································.167
8.3.5 Ψ-世界实验··············································································.167
8.4 问题表达和图像建模·······································································.168
8.4.1 用于分割的贝叶斯公式··································································.169
8.4.2 先验概率·················································································.169
8.4.3 灰度图像的似然·········································································.169
8.4.4 模型校准·················································································.171
8.4.5 彩色图像模型············································································.172
8.5 解空间分析···················································································.173
8.6 使用遍历马尔可夫链探索解空间························································.174
8.6.1 五类马尔可夫链动态过程·······························································.174
8.6.2 瓶颈问题·················································································.175
8.7 数据驱动方法················································································.176
8.7.1 方法一：原子空间中的聚类·····························································.176
8.7.2 方法二：边缘检测·······································································.180
8.8 计算重要性提议概率·······································································.180
8.9 计算多个不同的解··········································································.183
8.9.1 动机和数学原理·········································································.183
8.9.2 用于多种解的K-冒险家算法····························································.184
8.10 图像分割实验···············································································.185
8.11 应用：图像解析············································································.188
8.11.1 自上而下和自下而上的处理···························································.190
8.11.2 生成和判别方法········································································.190
8.11.3 马尔可夫链核和子核···································································.191
8.11.4 DDMCMC 和提议概率·································································.193
8.11.5 马尔可夫链子核········································································.200
8.11.6 图像解析实验···········································································.207
8.12 本章练习·····················································································.210
蒙特卡罗方法与人工智能
·X IV·
本章参考文献·······················································································.211
第9 章 哈密顿和朗之万蒙特卡罗方法····························································.215
9.1 引言····························································································.215
9.2 哈密顿力学···················································································.215
9.2.1 哈密顿方程···············································································.215
9.2.2 HMC 的简单模型········································································.216
9.3 哈密顿力学的性质··········································································.217
9.3.1 能量守恒·················································································.217
9.3.2 可逆性····················································································.218
9.3.3 辛结构和体积保持·······································································.219
9.4 哈密顿方程的蛙跳离散化·································································.220
9.4.1 欧拉方法·················································································.220
9.4.2 改良的欧拉方法·········································································.220
9.4.3 蛙跳积分器···············································································.221
9.4.4 蛙跳积分器的特性·······································································.222
9.5 哈密顿蒙特卡罗方法和朗之万蒙特卡罗方法·········································.223
9.5.1 HMC 建模················································································.223
9.5.2 HMC 算法················································································.224
9.5.3 LMC 算法················································································.226
9.5.4 HMC 调参················································································.228
9.5.5 HMC 的细致平衡证明···································································.229
9.6 黎曼流形HMC···············································································.230
9.6.1 HMC 中的线性变换·····································································.230
9.6.2 RMHMC 动力学·········································································.233
9.6.3 RMHMC 算法和变体····································································.235
9.6.4 RMHMC 中的协方差函数·······························································.236
9.7 HMC 实践·····················································································.237
9.7.1 受约束正态分布的模拟实验·····························································.237
9.7.2 使用RMHMC 对逻辑回归系数进行采样···············································.241
9.7.3 使用LMC 采样图像密度：FRAME、GRADE 和DeepFRAME ·······················.243
9.8 本章练习······················································································.248
本章参考文献·······················································································.249
第10 章 随机梯度学习················································································.250
10.1 引言···························································································.250
目 录
·XV·
10.2 随机梯度：动机和性质···································································.250
10.2.1 引例·····················································································.251
10.2.2 Robbins-Monro 定理····································································.253
10.2.3 随机梯度下降和朗之万方程···························································.254
10.3 马尔可夫随机场（MRF）模型的参数估计···········································.257
10.3.1 利用随机梯度学习FRAME 模型······················································.258
10.3.2 FRAME 的替代学习方法·······························································.259
10.3.3 FRAME 算法的四种变体·······························································.261
10.3.4 纹理分析实验···········································································.264
10.4 用神经网络学习图像模型································································.267
10.4.1 对比发散与持续对比发散······························································.267
10.4.2 使用深度网络学习图像的势能模型：DeepFRAME···································.268
10.4.3 生成器网络和交替反向传播···························································.271
10.4.4 协作网络和生成器模型································································.275
10.5 本章练习·····················································································.279
本章参考文献·······················································································.279
第11 章 可视化能级图················································································.282
11.1 引言···························································································.282
11.2 能级图的示例、结构和任务·····························································.282
11.2.1 基于能量的状态空间划分······························································.285
11.2.2 构造非连通图（DG）··································································.286
11.2.3 二维ELM 示例·········································································.287
11.2.4 表征学习任务的难度（或复杂度）····················································.289
11.3 广义Wang-Landau 算法··································································.290
11.3.1 GWL 映射的能垒估计··································································.291
11.3.2 用GWL 估算体积······································································.292
11.3.3 GWL 收敛性分析·······································································.294
11.4 GWL 实验···················································································.295
11.4.1 高斯混合模型的GWL 映射····························································.295
11.4.2 语法模型的GWL 映射·································································.301
11.5 用吸引-扩散可视化能级图······························································.305
11.5.1 亚稳定性和宏观划分···································································.306
11.5.2 吸引-扩散简介·········································································.307
11.5.3 吸引-扩散和Ising 模型································································.309
11.5.4 吸引-扩散ELM 算法（ADELM 算法）···············································.311
蒙特卡罗方法与人工智能
·X VI·
11.5.5 调优ADELM ···········································································.313
11.5.6 AD 能垒估计···········································································.314
11.6 用GWL 和ADELM 可视化SK 自旋玻璃模型······································.315
11.7 使用吸引?扩散可视化图像空间························································.318
11.7.1 图像星系的结构········································································.318
11.7.2 可视化实验·············································································.319
11.8 本章练习·····················································································.324
本章参考文献·······················································································.324

译 者 序
蒙特卡罗方法是一种基于概率统计和随机采样的数值计算方法。18 世纪70 年代，法
国数学家蒲丰用投针实验方法求圆周率π，一般被认为是蒙特卡罗方法的起源。20 世纪
30 年代，费米运用类似蒙特卡罗思路的方法来解决中子研究中的计算问题，但他并没有正
式命名或发表这些方法。直到20 世纪40 年代，为了解决科学实践中遇到的大规模数值计
算问题，冯·诺伊曼和乌拉姆等科学家正式提出并命名了蒙特卡罗方法。进入 21 世纪，
随着算力的高速增长和数据的极大丰富，蒙特卡罗方法在人工智能、计算机、工程学、物
理、生物、经济、金融等领域发挥着越来越重要的作用。
蒙特卡罗方法与人工智能任务的结合是一个自然而美妙的过程，因为两者都要面对并
解决“不确定性”问题。但蒙特卡罗方法的基础理论复杂，计算框架灵活，算法思路精
巧，导致很多研究者和工程师觉得用蒙特卡罗方法解决人工智能问题的门槛较高，领域内
也缺少系统探讨将蒙特卡罗方法与人工智能相结合的资料。
正因此，朱松纯教授和巴布·艾俊教授撰写了《蒙特卡罗方法与人工智能》（Monte
Carlo Methods），并于2020 年由施普林格（Springer）出版社出版。该书系统介绍了蒙特
卡罗方法及其在计算机和人工智能中的应用，包括计算机视觉、计算机图形学、机器学
习、机器人等领域，目的是在蒙特卡罗方法和人工智能之间架起一座桥梁，促进该方法更
好地应用于人工智能。
朱松纯教授和巴布·艾俊教授一直耕耘于统计学习和人工智能领域的前沿。自 20 世
纪 90 年代中期，朱松纯教授引领了视觉统计建模与计算理论的发展潮流，是数据驱动的
马尔可夫链蒙特卡罗（DDMCMC）、FRAME 等众多著名的统计学习方法的提出者。朱松
纯教授为人工智能的发展和人才培养做出了重要贡献。
《蒙特卡罗方法与人工智能》英文版完成后，朱松纯教授深感有必要推出中文版本，
以方便我国的学者特别是青年学生和初学者使用。我很荣幸受朱松纯教授委托负责本书中
文版的翻译出版。
感谢郑南宁教授和朱松纯教授给予的建议和帮助。感谢电子工业出版社李树林编辑，
他认真负责、精益求精，给出了大量修改建议。正是有了他们的帮助，本书的翻译、出版
才得以顺利完成。
特别说明，为了与原著对应，本书在翻译过程中对参考文献并未做修改，完全保留原
著格式；为了符合中文出版规范，书中修改了原著中的部分数学符号及表达形式。同时，
蒙特卡罗方法与人工智能
·IV ·
为了便于读者阅读，特准备了原著中的彩色图片资料包，读者可从电子工业出版社华信教
育资源网（https://www.hxedu.com.cn）下载。
译者虽尽心竭力以期尽可能传达原意，但因水平和能力有限，对此博大精深内容仅知
皮毛，书中难免有翻译谬误和不当之处，敬请读者批评指正，以期今后进一步修改完善。
魏平


中 文 版 序
在很多科学领域（如物理、化学与生物）与工程领域（如计算机视觉、图形学与仿
真），还有经济与社会模拟领域，蒙特卡罗方法已经得到广泛应用。近年来，统计建模与
机器学习成为人工智能发展的主要推动力量，而模型的模拟、推理、学习都是基于各种蒙
特卡罗方法的。可以说，统计建模与蒙特卡罗方法是未来人工智能科学发展的主流，这个
趋势在后面的发展中还会得到加强。
我在加州大学洛杉矶分校（UCLA）教了“蒙特卡罗方法”（Monte Carlo Methods）这
门课 10 多年后，在我的一名前博士生巴布·艾俊（Adrian Barbu）协助下根据教案于
2018 年完成了英文版教材的编写并出版。目前，我们正计划于 2024 年春天在北京大学开
设蒙特卡罗方法与人工智能的课程。
西安交通大学魏平教授用了几年的时间，把该教材翻译成中文，交由电子工业出版社
出版，这必将有助于促进蒙特卡罗方法在人工智能中的应用和推广。非常感谢魏平教授在
翻译和出版过程中所做的大量工作。由于中英文排版的习惯不同，蒙特卡罗方法术语众
多、公式复杂，他在翻译过程中做了很多细致的修改，几易其稿，多次校正，才形成了目
前的版本。
相信中文版一定有助于促进和推广蒙特卡罗方法在人工智能以及其他相关领域的应
用，助力我国人工智能人才的培养。
朱松纯
2023 年11 月于北京

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