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# Introduction: Jacob Westman

Hi, I'm a statistician/software developer, currently working at a small tech company in Sweden. My background is in statistics/mathematics and philosophy. I have been interested in category theory for a while without having the time to make a deep dive yet. I'm very curious about applications related to natural language processing and functional programming. I'm also curious to see if there are connection to statistics more generally. I've only seen a hand full of attempts at finding connections between these fields, but this is something I could see myself working a lot on if there is something interesting to find.

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edited April 2018

Hi, Jacob! There's bound to be interesting connections between statistics and category theory, because both disciplines are very general and conceptual. But discovering them will require a good knowledge of both fields. I'll give you a couple of references.

Comment Source:Hi, Jacob! There's bound to be interesting connections between statistics and category theory, because both disciplines are very general and conceptual. But discovering them will require a good knowledge of both fields. I'll give you a couple of references.
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edited April 2018

It's an interesting paper that introduces category theory to formalize the general concept of "statistical model", followed by discussions including a very interesting one that starts on page 1279, by Hans Brøns, who writes:

Peter McCullagh’s paper is exciting, because it can be seen as the start of a new, long overdue discussion of the mathematical foundation of the theory of statistics. He rightly points out that only part of the thinking in theoretical statistics is formalized mathematically and tries to extend the existing theory of statistical modelling using modern abstract mathematical tools. This is a great philosophical challenge, but it is a formidable pedagogical task to communicate the results to the statisticians.

The paper contains beyond the abstract definition of the new extended concept of a statistical model a treasure trove of examples and counterexamples, but I shall concentrate on an analysis of the definition of models. McCullagh’s idea is that parameter spaces and sample spaces which are usually treated as sets or measure spaces in applications have a richer structure defining what one could be tempted to call their “physical nature,” which should be reflected in the models and in the choice of transformations between them. This is done by giving them an inner structure, symmetry for example, and by considering each model as a unit in a greater universe of models. To give this a mathematical expression, the much loved and much hated theory of categories is used.

He then critiques and tries to improve McCullagh’s use of category theory. I don't know how much progress has been made since then, but it seems worthwhile to pursue this further.

Comment Source:Here's something worth reading: * Peter McCullagh, [What is a statistical model?](https://projecteuclid.org/euclid.aos/1035844977), _The Annals of Statistics_ **30** (2002), 1225-1310. It's an interesting paper that introduces category theory to formalize the general concept of "statistical model", followed by discussions including a very interesting one that starts on page 1279, by Hans Brøns, who writes: > Peter McCullagh’s paper is exciting, because it can be seen as the start of a new, long overdue discussion of the mathematical foundation of the theory of statistics. He rightly points out that only part of the thinking in theoretical statistics is formalized mathematically and tries to extend the existing theory of statistical modelling using modern abstract mathematical tools. This is a great philosophical challenge, but it is a formidable pedagogical task to communicate the results to the statisticians. > The paper contains beyond the abstract definition of the new extended concept of a statistical model a treasure trove of examples and counterexamples, but I shall concentrate on an analysis of the definition of models. McCullagh’s idea is that parameter spaces and sample spaces which are usually treated as sets or measure spaces in applications have a richer structure defining what one could be tempted to call their “physical nature,” which should be reflected in the models and in the choice of transformations between them. This is done by giving them an inner structure, symmetry for example, and by considering each model as a unit in a greater universe of models. **To give this a mathematical expression, the much loved and much hated theory of categories is used.** He then critiques and tries to improve McCullagh’s use of category theory. I don't know how much progress has been made since then, but it seems worthwhile to pursue this further. 
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Hi John, Thanks for that reference! This is exactly the sort of ideas I'm interested in pursuing. At the moment my level of understanding of category theory is to patchy to be able to really get a good grasp of what McCullagh is doing, but I'm really exited that this course can help me. Huge thanks for doing this!

Comment Source:Hi John, Thanks for that reference! This is exactly the sort of ideas I'm interested in pursuing. At the moment my level of understanding of category theory is to patchy to be able to really get a good grasp of what McCullagh is doing, but I'm really exited that this course can help me. Huge thanks for doing this!