7 ms·

Great article, especially considering how careless most "data scientists" are when they assume the arithmetic mean is going to be meaningful (for example, if yo

by qsort 6y ago

Great article, especially considering how careless most "data scientists" are when they assume the arithmetic mean is going to be meaningful (for example, if you are averaging frequencies, you want the harmonic mean; if you are averaging interest rates, you want the geometric mean, etc.) However, there are two big caveats: (1) You can't use any symmetric binary function, for the mean to make sense, f must be at least injective and monotonic (and you probably want it to be continuous as well). If you pick a suitable f, many useful properties of the mean are respected. (2) In general, for an arbitrary choice of f the resulting mean won't be homogenous, which is disqualifying for most practical applications (because, for example, the mean would not be guaranteed to have the same dimensions as the elements). Restricting ourselves to functions in the form $f(x) = x^a$ for $a \in \mathbb{R} \setminus 0$ yields a well-behaved collection of means (Holder means), guaranteed to be homogenous. As a bonus, for any set $S$ of positive real numbers, if we let $GM(S)$ be the geometric mean of $S$ and $M_{a}(S)$ be the mean of $S$ for some $a$, you can prove that $\lim_{a \rightarrow 0} M_{a}(S) = GM(S)$.

contravariant 6y ago
Those caveats are just the beginning. You generally also want the mean to be associative, you want the function they call 'g' to be invertible, and the harmonic mean and geometric mean have the simpler and more useful interpretation of taking the average of the reciprocal and logarithm respectively. All this article is doing, in a somewhat confused fashion, is trying to replace addition with another abelian (but not associative) operator. Which is fun, but ultimately not better than just assuming the arithmetic mean is going to be any help (which at least has some strong theorems that tell you it's going to converge to the expected value, usually).
- qsort 6y ago
  Yeah, I agree the author is abstracting so much that the object he's defining doesn't have any useful property left. I genuinely smiled when the article says "any function". "Any function" in math means any function :-) There's a reason why theorems involving arbitrary functions start with a list of qualifiers as hypotheses.
ttoinou 6y ago
> Great article, especially considering how careless most "data scientists" are when they assume the arithmetic mean is going to be meaningful That comes from the fact that they don't understand the data they are dealing with (as you said with dimensions). You cannot use the arithmetic mean on interest rates because it does not make sense to add them up. If you just go for the geometric mean "because it's a rate and I have read online I should use geometric mean" then you won't learn why you need to use the geometric mean. Frequencies also don't add up.... But wavelength or time do. So their inverse can be added. So their inverses can have an arithmetic mean. So you can physically meaningfully do the arithmetic mean of their inverse, and inverse the result back again
- qsort 6y ago
  > That comes from the fact that they don't understand the data they are dealing with (as you said with dimensions). You cannot use the arithmetic mean on interest rates because it does not make sense to add them up. If you're going down that route, which operations make sense and which don't boil down to the algebraic structure of the set of the possible values of the datatype. For example, you can't add dates but you can subtract dates, because dates have an affine structure, therefore the arithmetic mean of January 14 and October 13 is an operation that makes sense. Similarly with temperatures. The thing is that it's sometimes tricky and unintuitive to figure out what you're allowed to do: ratios are a good example because you can add them, it just doesn't mean what most people expect it to mean. I do agree that understanding your data goes a long way, but I believe making people more aware that tools other than "sum / n" exist is worthwhile, even though the article does have problems as I highlighted above.
  - ttoinou 6y ago
    Yes, but no temperatures cannot be substracted (nor added). For dates I believe you're right because they are vectors (what you think is "an absolute date in the calendar" is in fact the time elapsed after 00h00 year=1 after christ, so it's in fact relative)
    - 6y ago

by qsort 6y ago

contravariant 6y ago

Those caveats are just the beginning. You generally also want the mean to be associative, you want the function they call 'g' to be invertible, and the harmonic mean and geometric mean have the simpler and more useful interpretation of taking the average of the reciprocal and logarithm respectively. All this article is doing, in a somewhat confused fashion, is trying to replace addition with another abelian (but not associative) operator. Which is fun, but ultimately not better than just assuming the arithmetic mean is going to be any help (which at least has some strong theorems that tell you it's going to converge to the expected value, usually).

qsort 6y ago
Yeah, I agree the author is abstracting so much that the object he's defining doesn't have any useful property left. I genuinely smiled when the article says "any function". "Any function" in math means any function :-) There's a reason why theorems involving arbitrary functions start with a list of qualifiers as hypotheses.

ttoinou 6y ago

> Great article, especially considering how careless most "data scientists" are when they assume the arithmetic mean is going to be meaningful That comes from the fact that they don't understand the data they are dealing with (as you said with dimensions). You cannot use the arithmetic mean on interest rates because it does not make sense to add them up. If you just go for the geometric mean "because it's a rate and I have read online I should use geometric mean" then you won't learn why you need to use the geometric mean. Frequencies also don't add up.... But wavelength or time do. So their inverse can be added. So their inverses can have an arithmetic mean. So you can physically meaningfully do the arithmetic mean of their inverse, and inverse the result back again

qsort 6y ago
> That comes from the fact that they don't understand the data they are dealing with (as you said with dimensions). You cannot use the arithmetic mean on interest rates because it does not make sense to add them up. If you're going down that route, which operations make sense and which don't boil down to the algebraic structure of the set of the possible values of the datatype. For example, you can't add dates but you can subtract dates, because dates have an affine structure, therefore the arithmetic mean of January 14 and October 13 is an operation that makes sense. Similarly with temperatures. The thing is that it's sometimes tricky and unintuitive to figure out what you're allowed to do: ratios are a good example because you can add them, it just doesn't mean what most people expect it to mean. I do agree that understanding your data goes a long way, but I believe making people more aware that tools other than "sum / n" exist is worthwhile, even though the article does have problems as I highlighted above.
- ttoinou 6y ago
  Yes, but no temperatures cannot be substracted (nor added). For dates I believe you're right because they are vectors (what you think is "an absolute date in the calendar" is in fact the time elapsed after 00h00 year=1 after christ, so it's in fact relative)
  - 6y ago