蒙特卡罗方法与人工智能

　　本书全面叙述了蒙特卡罗方法，包括序贯蒙特卡罗方法、马尔可夫链蒙特卡罗方法基础、Metropolis算法及其变体、吉布斯采样器及其变体、聚类采样方法、马尔可夫链蒙特卡罗的收敛性分析、数据驱动的马尔可夫链蒙特卡罗方法、哈密顿和朗之万蒙特卡罗方法、随机梯度学习和可视化能级图等。为了便于学习，每章都包含了不同领域的代表性应用实例。本书旨在统计学和计算机科学之间架起一座桥梁以弥合它们之间的鸿沟，以便将其应用于计算机视觉、计算机图形学、机器学习、机器人学、人工智能等领域解决更广泛的问题，同时使这些领域的科学家和工程师们更容易地利用蒙特卡罗方法加强他们的研究。本书适合计算机、人工智能、机器人等领域的教师、学生阅读和参考，也适合相关领域的研究者和工业界的从业者阅读。

　　朱松纯，1996年获得哈佛大学计算机科学博士学位，现任北京通用人工智能研究院院长、北京大学人工智能研究院院长、北京大学讲席教授、清华大学基础科学讲席教授；曾任美国加州大学洛杉矶分校（UCLA）统计学与计算机科学教授，加州大学洛杉矶分校视觉、认知、学习与自主机器人中心主任。 他长期致力于为视觉和智能探寻一个统一的统计与计算框架：包括作为学习与推理的统一表达和数字蒙特卡罗方法的时空因果与或图（STC-AOG）。他在计算机视觉、统计学习、认知、人工智能和自主机器人领域发表了400多篇学术论文。他曾获得了多项荣誉，2003年因图像解析的工作成就获马尔奖，1999年因纹理建模、2007年因物体建模两次获得马尔奖提名。2001 年，他获得了NSF青年科学家奖、ONR青年研究员奖和斯隆奖。因为在视觉模式的概念化、建模、学习和推理的统一基础方面的贡献，他2008年获得了国际模式识别协会授予的J.K. Aggarwal奖。2013 年，他关于图像分割的论文获得了亥姆霍兹奖（Helmholtz Test-of-Time Award）。2017年，他因生命度建模工作获国际认知学会计算建模奖。2011年，他当选IEEE Fellow。他两次担任国际计算机视觉与模式识别大会（CVPR 2012，2019）主席。作为项目负责人，他领导了多个ONR MURI和DARPA团队，从事统一数学框架下的场景和事件理解以及认知机器人的工作。巴布·艾俊，2000 年获得俄亥俄州立大学数学博士学位，2005 年获得加州大学洛杉矶分校计算机科学博士学位（师从朱松纯博士）。2005年至2007年，他在西门子研究院从事医学成像研究工作，从开始担任研究科学家到后来升任项目经理。由于在边缘空间学习方面的工作成就，他与西门子的合作者获得了2011年Thomas A. Edison专利奖。2007年，他加入佛罗里达州立大学统计系，从助理教授到副教授，再到2019年担任教授。他发表了70多篇关于计算机视觉、机器学习和医学成像方面的论文，并拥有超过25项与医学成像和图像去噪相关的专利。魏平，西安交通大学人工智能学院教授、博士生导师，人工智能学院副院长，国家级青年人才，陕西高校青年创新团队（自主智能系统）带头人，西安交通大学“青年拔尖人才支持计划”A类入选者。西安交通大学学士、博士学位，美国加州大学洛杉矶分校（UCLA）博士后、联合培养博士。研究领域包括计算机视觉、机器学习、智能系统等。主持国家自然科学基金项目、国家重点研发计划子课题等科研项目十余项，作为骨干成员参与国家自然科学基金重大科学研究计划等课题多项。在TPAMI、CVPR、ICCV、ACM MM、AAAI、IJCAI等国际权威期刊和会议发表学术论文多篇，是十余个国际著名期刊和会议审稿人。担任中国自动化学会网联智能专委会副主任委员、中国图象图形学学会机器视觉专委会委员。

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·······························4