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The book that includes the results from these slides has broader scope, and also can be downloaded for free from arXiv: https://www.maths.ed.ac.uk/~tl/ed/ https

The book that includes the results from these slides has broader scope, and also can be downloaded for free from arXiv: https://www.maths.ed.ac.uk/~tl/ed/ https://www.maths.ed.ac.uk/~tl/ed/ "The starting point is the connection between diversity and entropy. We will discover: • how Shannon entropy, originally defined for communications engineering, can also be understood through biological diversity (Chapter 2); • how deformations of Shannon entropy express a spectrum of viewpoints on the meaning of biodiversity (Chapter 4); • how these deformations provably provide the only reasonable abundance-based measures of diversity (Chapter 7); • how to derive such results from characterization theorems for the power means, of which we prove several, some new (Chapters 5 and 9). Complementing the classical techniques of these proofs is a large-scale categorical programme, which has produced both new mathematics and new measures of diversity now used in scientific applications. For example, we will find: [...]" "The question of how to quantify diversity is far more mathematically profound than is generally appreciated. This book makes the case that the theory of diversity measurement is fertile soil for new mathematics, just as much as the neighbouring but far more thoroughly worked field of information theory"

contravariant 3y ago
Ah I was wondering about that. Their formula look suspiciously like the definition of Renyi entropy. I'm not too sure where the category theoretical stuff enters though. They mention that metric spaces have a magnitude, but their end result looks more like a channel capacity (with the confusion matrix being the probability to confuse one species with another). Which, you know, makes sense, if you've got 'N' signals but they're so easily confused with one another that you can only send 'n' signals worth of data then your channels are not too diverse. They do mention that this is equivalent to some modified version of the category theoretical heuristic, but is that really interesting? The link to Euler characteristic is intriguing, but from the way they end up at their final definition I'm not sure if metric spaces are really the natural context to talk about these things. It almost feels like they've stepped over an enriched category that would provide a more natural fit.
- autopoiesis 3y ago
  Metric spaces are enriched categories. They are enriched over the positive reals. The 'hom' between a pair of points is then simply a number: their distance.
  - drdeca 3y ago
    And, these non-negative real numbers, which are these homs, are “hom objects”, so regarded as objects in “the category with as objects the non-negative real numbers, and as morphisms, the ‘being greater than or equal to’ “ ? Is that right? So, I guess, (\R_{>= 0}, >=, +, 0) is like, a monoidal category with + as the monoidal operation? So like, for x,y,z in the metric space, the well, from hom(x,y) and hom(y,z) I guess the idea is there is a designated composition morphism from hom(x,y) monoidalProduct hom(y,z) to hom(x,z) which is specifically, hom(x,y)+hom(y,z) >= hom(x,z) (I said designated, but there is only the one, which is just the fact above.) I.e. d(x,y)+d(y,z) >= d(x,z) (Note: I didn’t manage to “just guess” this. I’ve seen it before, and was thinking it through as part of remembering how the idea worked. I am commenting this to both check my understanding in case I’m wrong, and to (assuming I’m remembering the idea correctly) provide an elaboration on what you said for anyone who might want more detail.)
    - consilient 3y ago
      > are “hom objects”, so regarded as objects in “the category with as objects the non-negative real numbers, and as morphisms, the ‘being greater than or equal to’ “ ? This works, but it's not quite what you want in most cases. There's a lot of stuff that requires you to enrich over a closed category, so instead we define `Hom(a,b)` to be `max(b - a, 0)` (which you can very roughly think of as replacing the mere proposition `a < b` with its "witnesses"). See https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf for more.
  - contravariant 3y ago
    Indeed they are. I'm saying it may not be the right context in this case. At least what they seem to be doing has little to do with metrics, and a lot more to do with probability distributions.
sesm 3y ago
Why not define ecological diversity as number of distinct biological species living in the area?
- autopoiesis 3y ago
  This is precisely the question answered by the OP. The answer is, "because there is a whole spectrum of things you might mean by 'diversity', of which 'number of distinct species' is only one extremum".
  - Scarblac 3y ago
    And also, I assume, because the concept of "species" isn't all that well defined?
    - sesm 3y ago
      It is well defined: a group of living organisms consisting of similar individuals capable of exchanging genes or interbreeding
      - LudwigNagasena 3y ago
        Ring species make your definition non-transitive. The same with species that can interbreed but exhibit hybrid breakdown.
      - sdenton4 3y ago
        I invite you to examine the notes of the international ornithological congress... The difference between species and subspecies is quite subtle, and subject to interpretation, because no one is really going to do the experiment to find out if two individuals of geographically district populations can actually still interbreed.
      - Scarblac 3y ago
        So if you have a few grams of soil and want to know how many species of micro organisms are in there, you're setting them up with dates to see which ones will end up breeding?

contravariant 3y ago

Ah I was wondering about that. Their formula look suspiciously like the definition of Renyi entropy. I'm not too sure where the category theoretical stuff enters though. They mention that metric spaces have a magnitude, but their end result looks more like a channel capacity (with the confusion matrix being the probability to confuse one species with another). Which, you know, makes sense, if you've got 'N' signals but they're so easily confused with one another that you can only send 'n' signals worth of data then your channels are not too diverse. They do mention that this is equivalent to some modified version of the category theoretical heuristic, but is that really interesting? The link to Euler characteristic is intriguing, but from the way they end up at their final definition I'm not sure if metric spaces are really the natural context to talk about these things. It almost feels like they've stepped over an enriched category that would provide a more natural fit.

autopoiesis 3y ago
Metric spaces are enriched categories. They are enriched over the positive reals. The 'hom' between a pair of points is then simply a number: their distance.
- drdeca 3y ago
  And, these non-negative real numbers, which are these homs, are “hom objects”, so regarded as objects in “the category with as objects the non-negative real numbers, and as morphisms, the ‘being greater than or equal to’ “ ? Is that right? So, I guess, (\R_{>= 0}, >=, +, 0) is like, a monoidal category with + as the monoidal operation? So like, for x,y,z in the metric space, the well, from hom(x,y) and hom(y,z) I guess the idea is there is a designated composition morphism from hom(x,y) monoidalProduct hom(y,z) to hom(x,z) which is specifically, hom(x,y)+hom(y,z) >= hom(x,z) (I said designated, but there is only the one, which is just the fact above.) I.e. d(x,y)+d(y,z) >= d(x,z) (Note: I didn’t manage to “just guess” this. I’ve seen it before, and was thinking it through as part of remembering how the idea worked. I am commenting this to both check my understanding in case I’m wrong, and to (assuming I’m remembering the idea correctly) provide an elaboration on what you said for anyone who might want more detail.)
  - consilient 3y ago
    > are “hom objects”, so regarded as objects in “the category with as objects the non-negative real numbers, and as morphisms, the ‘being greater than or equal to’ “ ? This works, but it's not quite what you want in most cases. There's a lot of stuff that requires you to enrich over a closed category, so instead we define `Hom(a,b)` to be `max(b - a, 0)` (which you can very roughly think of as replacing the mere proposition `a < b` with its "witnesses"). See https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf https://www.emis.de/journals/TAC/reprints/articles/1/tr1.pdf for more.
- contravariant 3y ago
  Indeed they are. I'm saying it may not be the right context in this case. At least what they seem to be doing has little to do with metrics, and a lot more to do with probability distributions.

sesm 3y ago

Why not define ecological diversity as number of distinct biological species living in the area?

autopoiesis 3y ago
This is precisely the question answered by the OP. The answer is, "because there is a whole spectrum of things you might mean by 'diversity', of which 'number of distinct species' is only one extremum".
- Scarblac 3y ago
  And also, I assume, because the concept of "species" isn't all that well defined?
  - sesm 3y ago
    It is well defined: a group of living organisms consisting of similar individuals capable of exchanging genes or interbreeding
    - LudwigNagasena 3y ago
      Ring species make your definition non-transitive. The same with species that can interbreed but exhibit hybrid breakdown.
    - sdenton4 3y ago
      I invite you to examine the notes of the international ornithological congress... The difference between species and subspecies is quite subtle, and subject to interpretation, because no one is really going to do the experiment to find out if two individuals of geographically district populations can actually still interbreed.
    - Scarblac 3y ago
      So if you have a few grams of soil and want to know how many species of micro organisms are in there, you're setting them up with dates to see which ones will end up breeding?