Gaussian Markov Random Fields: Theory and Applications
Gaussian Markov Random Fields: Theory and Applications
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高斯马尔可夫随机场：理论与应用
DOI：
10.1198/jasa.2006.s65
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发表时间：
2006
影响因子：
3.7
通讯作者：
Jun Yan
中科院分区：
文献类型：
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作者：
Jun Yan
Eric M. Vestrup的关于测量理论和集成的书是一本很好的读物。它是精心编写的，并给出了严格的，详细的治疗和一个特别清楚的介绍了经典和一些稍微不太标准的结果，衡量理论和整合。这本书开始了一个彻底的介绍设置系统，措施，扩展措施，勒贝格措施，和可衡量的功能。然后继续构造勒贝格积分，相对于勒贝格测度的积分，最后介绍L空间，Radon-Nikodym定理和测度空间的乘积。在众多的补充，这一标准的计划是结果的结构博雷尔集，勒贝格集，和哈迪-利特尔伍德定理。当然，已经有很多优秀的书籍涵盖了非常相似的范围：Halmos（1974），Rao（1987）和Billingsley（1995）仅举几例。尽管如此，我相信Vestrup的书是对该领域的一个有价值的补充，将成为许多研究人员和研究生的珍贵财产。作者自己的希望是为哈尔莫斯的经典文本提供一个更现代和更短暂的版本，这本书起源于20世纪50年代，因此与今天的人可能会寻找的方法和主题选择有些不同。在我看来，这是一个相当恰当的尝试。这本书是一个有点不太一般和不太抽象，但更详细的比最近的文本由饶（1987年），因为它的目的是要访问研究生刚刚开始学习措施和整合理论。最后，与Billingsley（1995）和其他一些经典著作不同，它只关注测度和积分，而不是仅仅把这个主题放在概率的背景下，当然，许多例子，练习和应用都来自概率。谁会想读这本书呢？作者建议将其作为一本教科书或背景阅读，为一个学期甚至一年的研究生水平课程的主题，给学生的数学，统计，甚至物理。在我看来，这将是一个很好的，如果雄心勃勃的选择作为教科书，它肯定会使伟大的背景阅读。第一次，甚至更先进的研究人员无疑会欣赏材料的深度和细节。没有步骤的证明是跳过或声称是微不足道的;一切都是奠定了读者在一个完整的，精确的，和耐心的方式。每一节都有丰富的练习，从简单的“手指练习”到更具挑战性的主题游览。虽然没有明确的解决方案给出的演习，他们是伴随着许多评论，把他们的背景下，以及相当详细的提示和证明大纲，许多更困难的问题。所有这一切使得这本书非常适合任何人想要通过自学学习测量和集成理论，只要他或她有很高的动机和兴趣深入学习这个主题。这是因为一些没有其他指导的初学者可能会对提供的大量信息和细节感到有点不知所措。这本书给出了一些建议，哪些部分不太重要，可以跳过。就算跳过了这些章节，也是一本很长的书。为了给出一个感觉，我只会说，这本书有近600页，其中密集挤满了小（和练习和例子，甚至更小）打印。这可能不会妨碍它作为教科书的实用性，因为讲座可以提供大纲和大画面，必须省略细节，然后Vestrup将以无可挑剔的方式提供。任何已经内化了基本概念的研究生或研究人员都会感谢这本书，因为它是一本优秀而细致的参考书，这可能最终成为这本书的主要用途。话虽如此，Vestrup的书将使一个有价值的参考书，不幸的是，该指数不是很广泛。我还应该提到，布局可以改进。一般来说，文本的结构应该更好一点，文本单元之间的空间稍微多一点。我怀疑，考虑到文本的长度，节省页面空间一直是优先考虑的（轻微）损害可读性。标题中的章节号有助于交叉引用，特别是因为“权利要求”和定理不带章节号。文本中偶尔也会出现一些错别字。所有这些相对次要的批评观点在第二版中可能会很容易地得到解决。总的来说，我非常喜欢阅读这本书。我喜欢Vestrup直观的解释和漂亮的，如果简短的，历史的帐户和见解，为什么一个路径或方法是采取另一个。我很欣赏他经常提到的替代术语和符号，这也是常用的。最重要的是，我对丰富的信息和大量完美的细节印象深刻。
Eric M. Vestrup’s book on measure theory and integration is an excellent read. It is carefully written and gives a rigorous, detailed treatment and a particularly clear presentation of the classical and some slightly less standard results in measure theory and integration. The book starts with a thorough introduction to set systems, measures, extensions of measures, Lebesgue measures, and measurable functions. It then continues to construct the Lebesgue integral, integrals relative to Lebesgue measure, and finally introduces L spaces, the Radon–Nikodym theorem, and products of measure spaces. Among the numerous additions to this standard program are results on the structure of Borel sets, Lebesgue sets, and the Hardy–Littlewood theorems. Of course, there is a host of excellent books already available that cover a very similar scope: Halmos (1974), Rao (1987), and Billingsley (1995) to name just a few. Nonetheless, I believe that Vestrup’s book is a valuable addition to the field and will be a prized possession for many researchers and graduate students. The author’s own hope is to provide a more modern and expository version of Halmos’ classic text, which has roots in the 1950s and thus has a somewhat different approach and choice of topics than one might look for today. In my opinion, it is a quite felicitous attempt at that. The book is a bit less general and less abstract but more detailed than the more recent text by Rao (1987), since it is intended to be accessible to graduate students just beginning to learn measure and integration theory. And finally, unlike Billingsley (1995) and a number of other classics, it focuses just on measure and integration, and does not solely put this topic in a probabilistic context, although of course, many examples, exercises, and applications are drawn from probability. So who might want to read this book? The author suggests to use it as a textbook or background reading for a semester or even year-long graduate level course on the topic addressed to students of mathematics, statistics, or even physics. In my opinion, it would be a good if ambitious choice as a textbook; it will certainly make for great background reading. The first-timer and even the more advanced researcher will no doubt appreciate that the material is presented in much depth and detail. No steps in the proofs are skipped or claimed to be trivial; everything is laid out for the reader in a complete, precise, and patient way. Each section comes with a wealth of exercises ranging from easy “finger exercises” to much more challenging topical excursions. Whereas no explicit solutions are given to exercises, they are accompanied by many comments that put them into context, as well as by quite detailed hints and proof outlines for many of the more difficult problems. All of this makes the book well suited for anybody wanting to learn measure and integration theory via self-study provided that he or she is highly motivated and interested to learn the subject in some depth. This is because some beginners with no other guidance might feel a bit overwhelmed with the amount of information and detail provided. The book gives some suggestions of which sections are less essential and may be skipped. Even if one skipped all of those sections, it would still be a long book. To give a sense of this, I will just say that the book has close to 600 pages which are densely packed with small (and for exercises and examples even smaller) print. This may not impede its usefulness as a textbook since lectures can provide the outline and the big picture and must necessarily leave out detail that Vestrup will then provide in impeccable fashion. Any graduate student or researcher who has internalized the basic concepts already will be thankful for this book as an excellent and meticulous reference, and this may become the book’s main use ultimately. Having said that Vestrup’s book will make a valued reference book, it is unfortunate that the index is not very extensive. I should also mention that the layout could be improved. In general, the text should have been a bit better structured, with slightly more space between text units. I suspect that given the length of the text, saving page space has been a priority to the (slight) detriment of readability. Section numbers in the header would aid cross referencing, particularly since “claims” and theorems do not carry section numbers. There are also some occasional typos in the text. All of these relatively minor points of criticism may be addressed quite easily in a second edition. Overall, I enjoyed reading this book very much. I liked Vestrup’s intuitive explanations and nice, if brief, historical accounts and insights into why one path or approach is taken over another. I appreciated his frequent references to alternative terminology and notation, which is also commonly used. And most of all, I was impressed with the wealth of information and the amount of flawless detail.