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I've been asked to contribute some lectures (or parts of lectures) to a course on "Mathematical Modelling", from a statistical perspective. This is to a rather mixed group of Mathematics undergraduates: some may have only seen one course on probability and statistics before.

There are a huge number of books with titles like "Introduction to Mathematical Modelling" (first example from a search) which typically concentrate upon the use of e.g. differential equations. Such sources often have a little on what I would call Monte Carlo simulation: adding some randomness to a model, and then running simulations. However, at least in the books I have looked at, there is extremely little which is "data driven".

  • I want to stress that such books are mathematically fairly unsophisticated, but at the same time, could not be called "popular science".

What I'm looking for is little case studies which start with some data, then discuss various probability models, then fit those models, and then make predictions or inferences, undertake some hypothesis testing (well, I much prefer a Bayesian point of view, but this is not how our students are taught) etc. However, what I've found is:

  • Any book with "statistical modelling" in the title seems much too advanced: e.g. starting off with generalised linear models
  • I like the approach taken by many Bayesian textbooks (to pick an example, Sivia and Skilling) but these tend still be rather sophisticated (relatively speaking). I guess I'm really after presentations in this style, but which assume less of the reader, without going so far as to be "popular science".
  • Of course, elementary stats textbooks have plenty of examples, but these tend to motivate things backwards from what I'm looking for: e.g. after introducing the Poisson distribution, there is some data presented (maybe just the sample mean given in fact), and some comments. But missing is seemingly why we might choose the Poisson distribution over other choices, etc.

I'd love some example sources which are in the style I'm after.

(I should also say that I'm hoping to escape actually giving lectures, but rather, give some sources to a colleague...)

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Not shure, whether this is worth an answer or just a comment, but I want the room this forum gives for answers only.

  1. Have look at this golf putting example, that Andrew Gelman gave in a webinar about stan. Forget about the Bayes-aspect of it. It just shows a standard model compared with an informed model and how the result improves when knowledge about the subject (not prior data!) is brought into model choice: https://youtu.be/T1gYvX5c2sM?t=2844 That should be worth noting, no matter of the mathematical knowledge.

  2. The second source is a youtube talk by Andrew Gelman as well. In his talk "crimes on data" he describes a number of published -yet very problematic- models. The examples are "real world" in the sense of "actually publshed, problems that you may be confronted with as a mathematician". You can learn a lot from that without lots of mathematics. I, for one, know, that small studies are unlikely to give significant results. It was not before this talk that I understood, how a significant result in a small study is even more of a problem. Start with this example, and if you like it, listen to the whole talk https://youtu.be/fc1hkFC2c1E?t=735

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  • $\begingroup$ I finally found time to watch the videos. (1) is actually really nice, and definitely what I was after, all be it probably too complicated. (I sort of see this as something to build up to, maybe). (2) is indeed a bit off topic, but it is excellent and I did indeed end up watching the whole thing. Thanks very much for sharing these links. $\endgroup$ – Matthew Daws Jun 18 '18 at 14:53
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But missing is seemingly why we might choose the Poisson distribution over other choices

This is indeed a frequent trait of probability textbooks – and even of research works in statistics. Somewhat of a taboo it seems. It was pointed out by the statistician A. P. Dawid (1982, § 4, p. 220):

Where do probability models come from? To judge by the resounding silence over this question on the part of most statisticians, it seems highly embarrassing. In general, the theoretician is happy to accept that his abstract probability triple $(\Omega, \mathcal{A}, \mathcal{P})$ was found under a gooseberry bush, while the applied statistician's model "just growed".

From a physicist's (or biologist's etc.) point of view this is a pity and a sin, because it doesn't help explaining why and how things happen. I even fear that today there's a tendency in the opposite direction: to see the probability model itself as an explanation. For example I've seen a couple of research articles stating that the explanation behind particular phenomena in economics was "luck" – which of course isn't an explanation at all, but just a declaration that we don't know the causes or mechanisms.

Well, sorry for this initial rant. It's really comforting to see someone asking why a Poisson model is used, rather than saying "because Poisson"!

Back to your question. It's difficult to find the kinds of examples you're asking about. You most likely will find them in physicists' writings; but their use of probability is often rather poor, and the maths may be too much for your purpose. The works of Jaynes's are an exception, so you might find examples there. Skim through his Probability Theory: The Logic of Science. In particular check §§ 6.10–6.11 for an example of how the Poisson distribution appears from physical reasoning.

An absolutely fantastic work where each probability model used is motivated by reasoning about the particular real context is Mosteller & Wallace's Inference in an authorship problem (1963). Really recommended. Maybe you can build mathematically simplified examples from it.

If you've found any useful examples in the literature, please do answer your own question and share them with us!

References

  • A. P. Dawid (1982): Intersubjective statistical models, in G. Koch, F. Spizzichino: Exchangeability in Probability and Statistics (North-Holland), pp. 217--232

  • E. T. Jaynes (2003): Probability Theory: The Logic of Science (Cambridge). Check out this link and this link

  • F. Mosteller, D. L. Wallace (1963): Inference in an authorship problem: A comparative study of discrimination methods applied to the authorship of the disputed Federalist papers, J. Am. Stat. Assoc. 58/302, pp. 275–309. Also at this link.

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