【00】Statistics and his mistresses
road
1
Statistics had sex with Algebraic Topology
Topological Data Analysis ∈ Statistics ∩ Algebraic Topology
Topological Data Analysis for Scientific Visualization by Julien Tierny
Julien Tierny: 9783319715063: Amazon.com: Books
作者：Edward Shi
链接：https://www.zhihu.com/question/276739747/answer/416928273
来源：知乎
著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。
2
Statistics had sex with Differential Geometry
Information Geometry ∈ Statistics ∩ Differential Geometry
Information Geometry and Its Applications by Shun-ichi Amari
作者：Edward Shi
链接：https://www.zhihu.com/question/276739747/answer/416928273
来源：知乎
著作权归作者所有。商业转载请联系作者获得授权，非商业转载请注明出处。
3
Statistics had sex with Group Representations
Algebraic Statistics
群表示论在概率统计中的应用
以下内容来自 http://www.mathsccnu.com/forum.php?mod=viewthread&tid=1516&extra=page%3D1 （代数统计 Algebraic Statistics）
Group Representations in Probability and Statistics (Lecture Notes Vol 11) Persi Diaconis
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Series: Lecture Notes Vol 11Paperback: 198 pages
Publisher: Inst of Mathematical Statistic (June 1988)
Language: English
ISBN-10: 0940600145
ISBN-13: 978-0940600140
djvu版本
Table of Contents
Preface v
Chapter 1 - Introduction
A. Introduction 1
B. Annotated bibliography 2
Chapter 2 - Basics of Representations and Characters
A. Definitions and examples 5
B. The basic theorems 7
C. Decomposition of the regular representation and Fourier 12
inversion
D. Number of irreducible representations 14
E. Products of Groups 16
Chapter 3 - Random Walks on Groups
A. Examples 17
B. The basic setup 21
C. Some explicit computations 25
D. Random transpositions: an introduction to the representation 36
theory of the symmetric group
E. The Markov chain connection 48
F. Random walks on homogeneous spaces and Gelfand pairs 51
G. Some references 61
H. First hitting times 64
Chapter 4 - Probabilistic Arguments
A. Introduction - strong uniform times 69
B. Examples of strong uniform times 72
C. A closer look at strong uniform times 75
D. An analysis of real riffle shuffles 77
E. Coupling 84
F. First hits and first time to cover all 87
G. Some open problems on random walk and strong uniform times 89
Chapter 5 - Examples of Data on Permutations and
Homogeneous Spaces
A. Permutation data 92
B. Partially ranked data 93
C. The ^-sphere Sd 99
D. Other groups 100
E. Statistics on groups 101
Chapter 6 - Metrics on Groups, and Their Statistical Uses
A. Applications of metrics 102
B. Some metrics on permutations 112
C. General constructions of metrics 119
D. Metrics on homogeneous spaces 124
E. Some Philosophy 129
Chapter 7 - Representation Theory of the Symmetric Group
A. Construction of the irreducible representations of 131
the symmetric group
B. More on representations of Sn 136
Chapter 8 - Spectral Analysis
A. Data on groups 141
B. Data on homogeneous spaces 147
C. Analysis of variance 153
D. Thoughts about spectral analysis 161
Chapter 9 - Models
A. Exponential families from representations 167
B. Data on spheres 170
C. Models for permutations and partially ranked data 172
D. Other models for ranked data 174
E. Theory and practical details 175
References 179
Index 193
另外，Benjamin Steinberg出版的Springer Universitext 系列图书 Representation Theory of Finite Groups中的最后一章也是讲这个。电子版见http://www.mathsccnu.com/forum.php?mod=viewthread&tid=1518
11 Probability and RandomWalks on Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
11.1 Probabilities on Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
11.2 RandomWalks on Finite Groups .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
11.3 Card Shuffling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
11.3.1 The Riffle Shuffle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
11.4 The Spectrum and the Upper Bound Lemma . . . . . . . . . . . . . . . . . . . . . . . 144
群表示论在统计中的应用是关于统计中的群论方法，这听起来不是怎么实际的东西。不过和魔术数学有关。群论是数学的重要分支，它对数学其它分支和物理、化学等其它科学有重要的应用。没想到和概率统计也有关。 下面是作者的回答。
我讲的课叫做“群表示论在统计中的应用”。讲课的第一天，我列出了20个能用语言表达清楚的应用问题。例如，要把一摞儿牌洗几次才能使其接近随机？并说明群论是已知的解决这些问题的唯一途径。出于解决实际问题的需要，我正在系统地研究群表示论，并将一个一个地处理上述问题。结果，你觉得群论更加优美了。正是这样。它在我思想中又活跃起来了。这是我的一个有趣的特点，我不能抽象地与数学打交道，我需要一个实际问题来思考数学。但是有了一个实际问题以后，我会学习一切能用来解放它的数学知识。我已经正规地学过至少30门纯数学课程，我尽得Ａ而且在学期末写出像样的文章，可这并不说明什么，这一点都不困难，我就是这样，我觉得有些人不会在用和问题中理解数学，不会实用地看待数学。我确实相信有一些这样的人，对他们来说，图表（diagram）和映射morphism）就是一切。我可不是这样，我得有应用。现在我在研究一个很优美的问题，需要深入了解元素为p-adic的２X２矩阵的表示理论方面的知识。这个问题来自捕捞大马哈鱼，这是学习p-adic的好办法。发现实际问题的艺术。http://blog.sina.com.cn/s/blog_88493ae90100sjzr.html
