So taking a step back for a second, I think the primary point of collaborative written or spoken communication is to take the picture or conceptual map in my head and put it in your head, as accurately as possible. Use of any terms should, in my view, be assessed against whether those terms are likely to create the right picture in a readerâs or listenerâs head. I appreciate this is a somewhat extreme position.
If everytime you use the term heavy-tailed (and itâs used a lotâa quick CTRL + F tells me itâs in the OP 25 times) I have to guess from context whether you mean the mathematical or commonsense definitions, itâs more difficult to parse what you actually mean in any given sentence. If someone is reading and doesnât even know that those definitions substantially differ, theyâll probably come away with bad conclusions.
This isnât a hypothetical corner caseâI keep seeing people come to bad (or at least unsupported) conclusions in exactly this way, while thinking that their reasoning is mathematically sound and thus nigh-incontrovertible. To quote myself above:
The above, in my opinion, highlights the folly of ever thinking âwell, log-normal distributions are heavy-tailed, and this should be log-normal because things got multiplied together, so the top 1% must be at least a few percent of the overall valueâ.
If I noticed that use of terms like âlinear growthâ or âexponential growthâ were similarly leading to bad conclusions, e.g. by being extrapolated too far beyond the range of data in the sample, I would be similarly opposed to their use. But I donât, so Iâm not.
If I noticed that engineers at firms I have worked for were obsessed with replacing exponential algorithms with polynomial algorithms because they are better in some limit case, but worse in the actual use cases, I would point this out and suggest they stop thinking in those terms. But this hasnât happened, so I havenât ever done so.
I do notice that use of the term heavy-tailed (as a binary) in EA, especially with reference to the log-normal distribution, is causing people to make claims about how we should expect this to be âa heavy-tailed distributionâ and how important it therefore is to attract the top 1%, and so...you get the idea.
Still, a full taboo is unrealistic and was intended as an aside; closer to âin my ideal worldâ or âthis is what I aim for my own writingâ, rather than a practical suggestion to others. As I said, I think the actual suggestions made in this summary are goodâreplacing the question âis this heavy-tailed or notâ with âhow heavy-tailed is thisâ should do the trick- and hope to see them become more widely adopted.
Iâm not sure how extreme your general take on communication is, and I think at least I have a fairly similar view.
I agree that the kind of practical experiences you mention can be a good reason to be more careful with the use of some mathematical concepts but not others. I think Iâve seen fewer instances of people making fallacious inferences based on something being log-normal, but if I had I think I might have arrived at similar aspirations as you regarding how to frame things.
(An invalid type of argument I have seen frequently is actually the âthings multiply, so we get a log-normalâ part. But as you have pointed out in your top-level comment, if we multiply a small number of thin-tailed and low-variance factors weâll get something thatâs not exactly a âparadigmatic exampleâ of a log-normal distribution even though we could reasonably approximate it with one. On the other hand, if the conditions of the âmultiplicative CLTâ arenât fulfilled we can easily get something with heavier tails than a log-normal. See also fn26 in our doc:
Weâve sometimes encountered the misconception that products of light-tailed factors always converge to a log-normal distribution. However, in fact, depending on the details the limit can also be another type of heavy-tailed distribution, such as a power law (see, e.g., Mitzenmacher 2004, sc. 5-7 for an accessible discussion and examples). Relevant details include whether there is a strictly positive minimum value beyond which products canât fall (ibid., sc. 5.1), random variation in the number of factors (ibid., sc. 7), and correlations between factors.
So taking a step back for a second, I think the primary point of collaborative written or spoken communication is to take the picture or conceptual map in my head and put it in your head, as accurately as possible. Use of any terms should, in my view, be assessed against whether those terms are likely to create the right picture in a readerâs or listenerâs head. I appreciate this is a somewhat extreme position.
If everytime you use the term heavy-tailed (and itâs used a lotâa quick CTRL + F tells me itâs in the OP 25 times) I have to guess from context whether you mean the mathematical or commonsense definitions, itâs more difficult to parse what you actually mean in any given sentence. If someone is reading and doesnât even know that those definitions substantially differ, theyâll probably come away with bad conclusions.
This isnât a hypothetical corner caseâI keep seeing people come to bad (or at least unsupported) conclusions in exactly this way, while thinking that their reasoning is mathematically sound and thus nigh-incontrovertible. To quote myself above:
If I noticed that use of terms like âlinear growthâ or âexponential growthâ were similarly leading to bad conclusions, e.g. by being extrapolated too far beyond the range of data in the sample, I would be similarly opposed to their use. But I donât, so Iâm not.
If I noticed that engineers at firms I have worked for were obsessed with replacing exponential algorithms with polynomial algorithms because they are better in some limit case, but worse in the actual use cases, I would point this out and suggest they stop thinking in those terms. But this hasnât happened, so I havenât ever done so.
I do notice that use of the term heavy-tailed (as a binary) in EA, especially with reference to the log-normal distribution, is causing people to make claims about how we should expect this to be âa heavy-tailed distributionâ and how important it therefore is to attract the top 1%, and so...you get the idea.
Still, a full taboo is unrealistic and was intended as an aside; closer to âin my ideal worldâ or âthis is what I aim for my own writingâ, rather than a practical suggestion to others. As I said, I think the actual suggestions made in this summary are goodâreplacing the question âis this heavy-tailed or notâ with âhow heavy-tailed is thisâ should do the trick- and hope to see them become more widely adopted.
Iâm not sure how extreme your general take on communication is, and I think at least I have a fairly similar view.
I agree that the kind of practical experiences you mention can be a good reason to be more careful with the use of some mathematical concepts but not others. I think Iâve seen fewer instances of people making fallacious inferences based on something being log-normal, but if I had I think I might have arrived at similar aspirations as you regarding how to frame things.
(An invalid type of argument I have seen frequently is actually the âthings multiply, so we get a log-normalâ part. But as you have pointed out in your top-level comment, if we multiply a small number of thin-tailed and low-variance factors weâll get something thatâs not exactly a âparadigmatic exampleâ of a log-normal distribution even though we could reasonably approximate it with one. On the other hand, if the conditions of the âmultiplicative CLTâ arenât fulfilled we can easily get something with heavier tails than a log-normal. See also fn26 in our doc:
)