Firstly, I suppose that not all active members of this interesting site are statisticians as their job. Otherwise the question being asked as follows does not make any sense! I respect them of course, but I need an explanation that is a bit more practical rather than conceptual.

I start with an example from Wikipedia to define point process:

Let S be locally compact second countable Hausdorff space equipped with its Borel σ-algebra B(S). Write $\mathfrak{N}$ for the set of locally finite counting measures on S and $\mathcal{N}$ for the smallest σ-algebra on $\mathfrak{N}$ that renders all the point counts ... measurable.

To me this has no meaning. An explanation in an engineering context is more understandable to me.

Comment: Most of time I found Wikipedia's explanations useless due to similar complicated text (at least to me). From my experience there are only two types of reference books for statistics: a) extremely simplified b) extremely complicated!
Reading both has no benefits for me at all!


  • Do you have a solution for this problem? Or similar experience?

For those who found this post useful there are benefits to check also: References for consulting statisticians to offer their clients which discusses a related topic from different perspective.

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    $\begingroup$ +1 Good question. Your pain is shared by all who make an effort to learn such concepts. Terrible references increase the friction and effort to learn, rather than decrease it. $\endgroup$ – Iterator Oct 22 '11 at 12:44
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    $\begingroup$ (+1) I'm wondering if this question has a "right" answer, though. Should it be CW? $\endgroup$ – cardinal Oct 22 '11 at 13:14
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    $\begingroup$ @cardinal I'm not sure. On SO, I'd say that CW applies, but I'm not sure of the standards for CW-ish questions on CV. There are a lot of questions here that cannot be answered with a singular right answer. If this is made into CW, then it may be useful to help edit the question a little (and just a little!) to focus it a little on how to assist someone with a desire to learn, but without local references and pedagogical support. $\endgroup$ – Iterator Oct 22 '11 at 13:31
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    $\begingroup$ @Developer: While I appreciate the accept, I think it is a bit premature. Maybe you can unaccept, to encourage others to post answers? I'd really like to know some of the answers from others, too. :) $\endgroup$ – Iterator Oct 22 '11 at 13:32
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    $\begingroup$ I'd interpret these as "overly simplified, with little practical ability to extend to practical use" and "overly abstract and detailed, and not demonstrably useful for practical purposes." This might be akin to describing OLS linear regression as "fitting a line to data" (overly simplified) or, see this example for a presentation that is precise, but not really practical. $\endgroup$ – Iterator Oct 24 '11 at 20:01

If I might clarify, your question appears to be: "What can I use to understand mathematics if a major resource like Wikipedia makes no sense?" Keep in mind that even a person who has mastered a concept had to begin with a period of not understanding it, and then go through a learning process, albeit one that almost never involved learning much from Wikipedia.

Having spent a lot of time studying things that are described quite atrociously on Wikipedia, I can assure you that even when one understands the concepts quite well, it is difficult to make sense of what was going through the minds of one or more authors/editors on Wikipedia. It is not uncommon to see mathematical and statistical concepts mutilated by a bunch of people with a very rough grasp of the concepts or in pursuit of advancing yet another field's weak grasp of the fundamental concept. (I would say more, but it is hard to do so without sounding unduly pessimistic about the efforts of Wikipedians, especially those from certain other disciplines.)

On a more constructive note, the best references are typically those textbooks edited by publishers with a strong track record of editing and publishing good works in the given field. Authors and editors in such cases have a reputation among their peers for the quality of their scholarship and rigor, and a series of successive editions usually indicates acceptance by other teachers and researchers.

There are many levels of quality between that level and Wikipedia. If the print editions are not available, using Amazon's "Search inside the book" or Google Books may be the best alternatives.

For other web-accessible references, you may find that review articles or manuals for non-specialist practitioners are most useful. An example of this is the statistics handbook published by NIST.

You may need to synthesize your own understanding by way of looking for articles in Google Scholar. For instance you could query ["a point process is a"] and examine the definitions offered in various articles. Alternatively, a web search like ["point process" pdf site:edu] will turn up lecture notes, slides, and tutorials. The first result for that query appears to be "An introduction to point processes". The key idea is that one should search for terms that either tend to appear or may appear in the appropriate level of material that would define and introduce the concept, whether or not the phrasing was intended to denote that the reference has some relevant exposition (e.g. a journal article may define something in a useful way, even if it isn't intended to be an introductory text).

It is impossible to push against the bad edits on Wikipedia: for certain articles, the number of bad editors exceeds the number of people who can tolerate fixing their errors.

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    $\begingroup$ This answer is really useful. I appreciate your supportive, motivational and kind explanation and also yours tips-and-tricks for googling. $\endgroup$ – Developer Oct 22 '11 at 13:24
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    $\begingroup$ No problem! Your time in your pursuit should be well spent, so you don't lose interest in statistics. :) $\endgroup$ – Iterator Oct 22 '11 at 13:26
  • $\begingroup$ Also, I also tried stuff like ["a point process is" "in practice"], but that turned out to be useless. It's good to observe what words are common in phrasing definitions and examples, and then search for those phrases. $\endgroup$ – Iterator Oct 22 '11 at 13:28
  • $\begingroup$ What exactly is "useless" about these searches? The top hits on Googling "point process" turn up a broad range, from the very simplified through the precisely mathematical. $\endgroup$ – whuber Oct 24 '11 at 18:59
  • $\begingroup$ @whuber Apologies: that prior statement is not reproducible. Google customizes results, and at that time some of the other queries I tried (e.g. ["a point process is" "in practice"], which was an experiment to get practical, hands-on insights) were not very informative. In time, I'll delete the previous comment. It doesn't serve any purpose when it's not reproducible. $\endgroup$ – Iterator Oct 24 '11 at 19:46

I understand where you are coming from. In my field of psychology, there are a lot of resources that present statistics in a superficial way. This is fine for many students, yet such books do not provide the prerequisites for reading more sophisticated books.

It sounds like you need to (a) get a better picture of the range of statistics books out there and the necessary prerequisites that different resources imply. (b) define your learning goals; (c) identify your current knowledge; and (d) put it all together to create a learning environment.

A. Develop sense of the statistics resources landscape

Perhaps this provides a rough sense of the introductory statistics resource landscape organised on a continuum of rigour and mathematical sophistication.

  • Cookbooks: Some resources have a cookbook style, showing how to use software and providing tips on when to use and how to interpret statistical output (e.g., SPSS Survival Manual). These books serve a purpose for people who have standardised data analysis needs, and don't have the time nor the inclination to engage in deeper learning.
  • Standard introductions to statistics: There's also a wide range of introductory statistics books that vary in the degree of mathemtical rigour. For some the only prerequisite is that you can perform basic algebra. HyperStat provides one online example.
  • More sophisticated introductions to statistics: While arguably on a continuum, other introductions to statistics are a little more mathematically rigorous. It seems to me that a big difference is whether the textbook assumes the reader is familiar with calculus and linear algebra. As @iterator has noted, the online Handbook of Engineering Statistics is one example of such a more sophisticated resource.
  • Mathematical Statistics: At the next level of rigour are resources that one might classify as mathematical statistics. Check out for example Virtual Laboratories in Probability and Statistics, course notes for this MIT subject on mathematical statistics, or some of the video courses here on mathematical statistics.

B. Define your learning goals

What is it that you want to do with this statistics knowledge? How important is mathematical rigour? Do you need to understand mathematically sophisticated descriptions that might appear on Wikipedia?

C. Identify your current knowledge

For many students in the social sciences, engaging with mathematically sophisticated textbooks effectively requires learning or refreshing a large amount of mathematics. However, if you have an engineering background, then I imagine that engaging in a more mathematical treatment should not be a major issue.

D. Put it all together

Once you've defined what you want to learn, what you already know, and the prerequisites required to learn the new material, the challenge is to find the best resources for you.

  • Does the resource need to be freely available on the Internet, or are you willing to buy or borrow a book?
  • Are you able to learn sufficiently from a textbook style resource or do you want or need a classroom environment with a lecturer verbalising the material and the structure that a course provides?
  • If you are after a textbook on a particular, what textbook do you find clearest, best, and so on?

Once you have answers to the above questions, you may have more specific questions that would be suited to this site. E.g., "I know x, y, z, and what is a good textbook that explains a, b, c?"

  • $\begingroup$ +1 There are useful hints in this answer too. Your comment is a bit general so although does not fully describe my situation but provides interesting tips to deal with difficulties in learning statistics. The level of interest in the question is advanced statistics (although some textbooks with a title including "advanced" are introductory to me) and the problem appears is the lack of suitable references for readers as non-statistics scholars. This could be helpful for my other post. Thanks $\endgroup$ – Developer Oct 26 '11 at 1:10
  • $\begingroup$ @Developer I guess your question was fairly general, so I thought a general response would be useful. If you have something specific that you want to learn, then you might want to ask a specific question on that: e.g., a specific question on text books that explain point processes, or a question asking for specific exposition of an idea that you don't understand. $\endgroup$ – Jeromy Anglim Oct 26 '11 at 1:39
  • $\begingroup$ I absolutely appreciate your great answer. As I wrote it was in some key points very useful to me and I'm sure others with the similar concern will find it even more helpful. Your comment brought to my attention that it might be sometime I just think that I know something. This answer and comment encourage me to establish a structured strategy to solve the problem and I recommend it to be considered important by others too. $\endgroup$ – Developer Oct 26 '11 at 4:05
  • $\begingroup$ Nice answer! This is good advice when starting a reading project. I like the structure and generality, as it applies to learning in other question fields, too. $\endgroup$ – Iterator Oct 30 '11 at 12:08

Just to add to excellent answer given by Iterator. Sometimes it is not necessary to understand the concept to successfully use it. I often encounter unknown concepts when reading articles, but before trying to figure out what they mean in external source, I always check whether it is possible to understand what is going on if I assume that the unknown concept is just a new fancy name for something that I already know. More often than not, only some specific easily understandable property of that new concept is used, so I in the end I understand what the author of the article did, and I can decide whether it is useful or not.

Taking the example definition in your question, it is possible to simplify it, and actually you can find the simplifications in the same wikipedia article. For example $S=\mathbb{R}^d$ is locally compact second countable Hausdorff space, so if you are working only in $\mathbb{R}^d$, which is nice and understandable, just look for examples where $S=\mathbb{R}^d$ and ignore everything else. Unknown concept can become very simple for your particular problem.

Note that this approach does not always work. Sometimes you really need to go deep, and then wikipedia is as good as the starting point for the search. In this case nothing beats a good book. Sometimes it is very easy to find one, sometimes unfortunately there are none.

  • $\begingroup$ Your point seems interesting and helpful to me and is worth to be kept in mind. However as an engineer I always suppose there should be an important meaning/concept/assumption/criteria/... in the use every word. So it could be mislead me if I simplify (well I'm not a well-educated simplifier). It seems to me that sometime a mathematical/statistical statement of a clear verbal statement/situation/problem makes it too complicated. $\endgroup$ – Developer Oct 25 '11 at 9:36
  • $\begingroup$ I'll second the suggestions that usage can make a concept clearer, even though its original definition may not yet make sense. A very good example of this is from Euclid's Elements. It looks trivial when you first begin to read it, if you've already studied number theory, geometry, algebra, etc. However, as you progress, it is a marvel of simplicity and clarity. I frequently tried to reduce the number of steps needed for a proof, and I don't recall that I ever succeeded. The process of reading this book readies the mind for reading and writing other definitions. $\endgroup$ – Iterator Oct 25 '11 at 14:49

I think the problem exists but that you are overstating it. If you are persistent in your searching you will find extremely useful books and other sources that keep to a middle ground between the extremely technical (e.g., most articles in the Journal of the American Statistical Association; most pieces written by Andrew Gelman, Bradley Efron, or Donald Rubin) and the extremely simple. I've spent quite a bit of time searching for these 'middle-ground' sources myself. If you care to see some of my recommendations you'll find a set of them at yellowbrickstats.com. I also often find useful information at David Garson's site at North Carolina State U.

  • $\begingroup$ Well, one of the reasons of posting this question is encouraging people like you to share their experiences with the community. Thanks for the links and sharing your experience. $\endgroup$ – Developer Oct 27 '11 at 1:42

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