3 ms·

"Time average" and "ensemble average" are standard vocabulary in the statistical mechanics literature. Your comment is essentially a restatement of the article'

"Time average" and "ensemble average" are standard vocabulary in the statistical mechanics literature. Your comment is essentially a restatement of the article's point. I think it's uncharitable to say the article would be easier to understand if it didn't use the language of ergodicity. Its explicit goal is to show how non-ergodicity leads to an example like yours. So of course your comment seems easier to understand. But that's because you're just saying different distributions can be parameterized by the same mean. Ergodicity is about a lot more than that, and the language of ergodicity was the entire exercise here.

kgwgk 6y ago
> "Time average" and "ensemble average" are standard vocabulary in the statistical mechanics literature But their application to non-standard-mechanical things is very confusing. Of course wealth is not ergodic. Ergodicity would mean that the distribution is always the same. Every point in time would be identical to every other point in time and growth would be impossible.
- fractionalhare 6y ago
  I agree it's not perfectly explained. But I think someone new to ergodic theory would find the article clearer (or at least more helpful overall) than your second paragraph here.
  - kgwgk 6y ago
    “What we're seeing is that even though the expected value is positive, and the ensemble average is increasing, the time average for any single person is usually decreasing. The average of the entire "system" increases, but that doesn't mean that the average of a single unit is increasing.” Someone new to ergodic theory may understand from that article that if wealth was ergodic the average for every trajectory would increase like the average for the entire system. But that doesn’t make sense.
loup-vaillant 6y ago
Thing is, I don't believe we even care about the time average. What we care about is the evolution of the distribution of outcomes over time. More specifically: - The distribution of outcomes at certain points of interest in time (like the valuation of my company when I intend to sell it). - The probability that we cross a catastrophic threshold at some point (like bankruptcy). Time average is a terrible metric to estimate those things. Heck, I'm not sure it can measure anything of interest, besides our own mistaken intuitions. It should probably be called something like "time average fallacy".
- fractionalhare 6y ago
  I'm a little confused - ergodic theory very much cares about the time average. Or do you mean the toy example of betting shouldn't care about it? It seems like you think the problem here is too unsophisticated for ergodic theory or something. Which, fine sure. But this isn't an article intended to teach you about betting. It's an article intended to teach you about ergodicity, using betting as a toy example. The author isn't trying to introduce the best way to analyze betting strategies, they're trying to show what non-ergodicity is. And I think they basically succeed. Just meet the article where it is, for its intended usage.
  - loup-vaillant 6y ago
    This is not about the example. What I'm saying that no betting at all should care about the time average. Betting is about having good estimation of outcomes, and time averages only helps you when the process is ergotic. That's a very special case. For everything else (that is, non-ergotic processes), your time average is crap, and you must look at the distribution of outcomes directly. Even the ensemble average is not enough. Averages are crap at visualising skewed distributions. For those you want the median, the quartiles, sometimes even the percentiles. --- To be honest, this "ergotic theory" shows signs of snake oil. The definition of ergodicity itself is dead simple, so it's pretty easy to evaluate. What seems pretty clear is that ergodic processes are the exception. And a pretty uninteresting one at that, since it's a class of processes that people will have good intuitions about. It would then seem that ergodic theory is more interested in the non ergodic processes (the very point of this blog post is to warn us about them). That is, processes that lack some property —the general case. And surprise, since the time average and ensemble averages are different, and you only care about the ensemble average (well, the ensemble distribution really), the time average won't help you. Be afraid, or lose your assets. That's why I see snake oil: what works on non-ergodic processes will also work on the ergodic ones. Unless you need to make a split second decision using your intuition (which while inadvisable is safer with ergodic processes), there's no need to make the distinction at all. Just analyse your process without without assuming it will be ergodic, the results will be applicable even if it is.
Ericson2314 6y ago
Yes, thank you. This intro doesn't get to the depths of the issue. https://www.nature.com/articles/s41567-019-0732-0 https://www.nature.com/articles/s41567-019-0732-0, by one of the pioneers of the "egondocity economics" is very nice for both going over the math and the academic history of the error. Given the illustrious history of statistical mechanics into Modern probability theory, information theory, theoretical computer science, etc., it's a real shame Econonomics is still stuck with this bad math. https://aeon.co/ideas/how-ergodicity-reimagines-economics-for-the-benefit-of-us-all https://aeon.co/ideas/how-ergodicity-reimagines-economics-fo... the pop-sci narrative here really doesn't seem that much an exaggeration. The way non-ergonomics fixes the math and confirms some real-world intuitions is quite profound. And certainly there is a lot to critique with orthodox economics' math. (See https://themountaingoateconomics.com/ https://themountaingoateconomics.com/ for another example.)
- kgwgk 6y ago
  What is a shame is how “egondocity economics” (maybe a reference to the size of Peters’ ego?) misrepresents some things. Are you calling “bad math” the expected utility theory developed by von Neumann (et al.)? He knew one thing or two about ergodicity, information theory, computer science, etc.
  - Ericson2314 6y ago
    What is being mispreresented? I read https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenster..., And there's no notion of time let alone non-ergoticity in the formula. I am not familiar of with the rest of its book, but I wouldn't be surprise if it's similarly fine, building a theory similarly of rich theorems about very simple models. If so, the problem isn't Von Neumann's math then, even if the general aim of the endever was misinspired by Bernoulli's primitive notions. The problem would be all the math cargo culters in economics who constantly try to the premise premises of math theorems as if they were broad social laws. I mean don't get me wrong, I am no fan of Von Neumannn's politics, but obviously I am not going to fight his pure math.

kgwgk 6y ago

> "Time average" and "ensemble average" are standard vocabulary in the statistical mechanics literature But their application to non-standard-mechanical things is very confusing. Of course wealth is not ergodic. Ergodicity would mean that the distribution is always the same. Every point in time would be identical to every other point in time and growth would be impossible.

fractionalhare 6y ago
I agree it's not perfectly explained. But I think someone new to ergodic theory would find the article clearer (or at least more helpful overall) than your second paragraph here.
- kgwgk 6y ago
  “What we're seeing is that even though the expected value is positive, and the ensemble average is increasing, the time average for any single person is usually decreasing. The average of the entire "system" increases, but that doesn't mean that the average of a single unit is increasing.” Someone new to ergodic theory may understand from that article that if wealth was ergodic the average for every trajectory would increase like the average for the entire system. But that doesn’t make sense.

loup-vaillant 6y ago

Thing is, I don't believe we even care about the time average. What we care about is the evolution of the distribution of outcomes over time. More specifically: - The distribution of outcomes at certain points of interest in time (like the valuation of my company when I intend to sell it). - The probability that we cross a catastrophic threshold at some point (like bankruptcy). Time average is a terrible metric to estimate those things. Heck, I'm not sure it can measure anything of interest, besides our own mistaken intuitions. It should probably be called something like "time average fallacy".

fractionalhare 6y ago
I'm a little confused - ergodic theory very much cares about the time average. Or do you mean the toy example of betting shouldn't care about it? It seems like you think the problem here is too unsophisticated for ergodic theory or something. Which, fine sure. But this isn't an article intended to teach you about betting. It's an article intended to teach you about ergodicity, using betting as a toy example. The author isn't trying to introduce the best way to analyze betting strategies, they're trying to show what non-ergodicity is. And I think they basically succeed. Just meet the article where it is, for its intended usage.
- loup-vaillant 6y ago
  This is not about the example. What I'm saying that no betting at all should care about the time average. Betting is about having good estimation of outcomes, and time averages only helps you when the process is ergotic. That's a very special case. For everything else (that is, non-ergotic processes), your time average is crap, and you must look at the distribution of outcomes directly. Even the ensemble average is not enough. Averages are crap at visualising skewed distributions. For those you want the median, the quartiles, sometimes even the percentiles. --- To be honest, this "ergotic theory" shows signs of snake oil. The definition of ergodicity itself is dead simple, so it's pretty easy to evaluate. What seems pretty clear is that ergodic processes are the exception. And a pretty uninteresting one at that, since it's a class of processes that people will have good intuitions about. It would then seem that ergodic theory is more interested in the non ergodic processes (the very point of this blog post is to warn us about them). That is, processes that lack some property —the general case. And surprise, since the time average and ensemble averages are different, and you only care about the ensemble average (well, the ensemble distribution really), the time average won't help you. Be afraid, or lose your assets. That's why I see snake oil: what works on non-ergodic processes will also work on the ergodic ones. Unless you need to make a split second decision using your intuition (which while inadvisable is safer with ergodic processes), there's no need to make the distinction at all. Just analyse your process without without assuming it will be ergodic, the results will be applicable even if it is.

Ericson2314 6y ago

Yes, thank you. This intro doesn't get to the depths of the issue. https://www.nature.com/articles/s41567-019-0732-0 https://www.nature.com/articles/s41567-019-0732-0, by one of the pioneers of the "egondocity economics" is very nice for both going over the math and the academic history of the error. Given the illustrious history of statistical mechanics into Modern probability theory, information theory, theoretical computer science, etc., it's a real shame Econonomics is still stuck with this bad math. https://aeon.co/ideas/how-ergodicity-reimagines-economics-for-the-benefit-of-us-all https://aeon.co/ideas/how-ergodicity-reimagines-economics-fo... the pop-sci narrative here really doesn't seem that much an exaggeration. The way non-ergonomics fixes the math and confirms some real-world intuitions is quite profound. And certainly there is a lot to critique with orthodox economics' math. (See https://themountaingoateconomics.com/ https://themountaingoateconomics.com/ for another example.)

kgwgk 6y ago
What is a shame is how “egondocity economics” (maybe a reference to the size of Peters’ ego?) misrepresents some things. Are you calling “bad math” the expected utility theory developed by von Neumann (et al.)? He knew one thing or two about ergodicity, information theory, computer science, etc.
- Ericson2314 6y ago
  What is being mispreresented? I read https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenstern_utility_theorem https://en.wikipedia.org/wiki/Von_Neumann%E2%80%93Morgenster..., And there's no notion of time let alone non-ergoticity in the formula. I am not familiar of with the rest of its book, but I wouldn't be surprise if it's similarly fine, building a theory similarly of rich theorems about very simple models. If so, the problem isn't Von Neumann's math then, even if the general aim of the endever was misinspired by Bernoulli's primitive notions. The problem would be all the math cargo culters in economics who constantly try to the premise premises of math theorems as if they were broad social laws. I mean don't get me wrong, I am no fan of Von Neumannn's politics, but obviously I am not going to fight his pure math.