蒙特卡罗方法与人工智能

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