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What is ergodicity? (2016)
- uniformlyrandom 7y agoInterestingly, ergodicity has two meanings (https://www.merriam-webster.com/dictionary/ergodic https://www.merriam-webster.com/dictionary/ergodic): 1. of or relating to a process in which every sequence or sizable sample is equally representative of the whole (as in regard to a statistical parameter) 2. involving or relating to the probability that any state will recur especially : having zero probability that any state will never recur Nassim Taleb mostly uses ergodicity in the second meaning, while the article concentrates on the first. I find it utterly confusing.
- pcstl 7y agoHonest question: Aren't both equivalent? If the process will eventually traverse its entire state space, you should be able to infer the behavior of the process across all time from a finite (sufficiently large) sample of the process's behavior.
- AstralStorm 7y agoSecond version is much more loose, e.g. most semi-stationary processes fit the bill. In fact article makes a mistake by requiring stationary nature from ergodic process. You can have an ergodic process of the first kind where ML expectation is equal to the value of the process function, which nonetheless is not stationary in absolute sense. (Because definite integral does not match indefinite.) Just perfectly predictable and having the indefinite integral identical to integral over space state. Which implies strictly bounded but not exactly stationary in any defined term. Likewise you can prove an equivalence class of the process to be identical to another known ergodic process without actually calculating such difficult to obtain complete integrals. And ergodicity is not required for representative sampling either, just definite error bounds on predictive (statistical) power or likelihood estimates. Which can be estimated or directly calculated for many kinds of processes accurately. It is only required for certainty.
- Simon_says 7y agoYes, that's the content of the Birkhoff Ergodic Theorem.
- Retra 7y agoThe dictionary is not a reliable place to look up technical terms. (Actually, it's not a reliable place to look up most anything.)
- uniformlyrandom 7y agoAre you contesting either of the meanings?
- Retra 7y agoNo, I'm questioning the validity of answering the title question by consulting a dictionary. That's not how one should go about answering technical questions, even if we find it works sometimes. (The proper way to address the question would also allow you to recognize when a dictionary was wrong.) Dictionaries don't actually tell you what a word means, they give examples of what it may mean from common usage.
- vbuwivbiu 7y agoTaleb uses his own definition of ergodicity which he's yet to state
- dxbydt 7y agoI’m curious why you would think that. afaik, Taleb has repeatedly & explicitly stated “no probability without ergodicity”, which is pretty much a frequentist mantra. The problem with non-statisticians like Nate Silvers (who holds a BS in Econ, as opposed to Taleb who earned his PhD in Statistics), is that they traffic in electoral polls involving human subjects which are anything but ergodicity. Yet, they brazenly attribute probability models to their conclusions. That has severe definitional issues, which is what Taleb objects to. As to the gatekeeping issues, my mom buys eggs for $3 and groceries for $7. 3+7=10, so she hands over a $10 bill. Wait a minute, does that mean my mom is a mathematician, because she has engaged in arithmetic, which is math ? That’s the issue here. Silvers can simply post how many people want to vote which way & leave it at that. It’s just an opinion poll. Instead he engages in an elaborate charade where he takes polls of polls, ensemble averages, then says this is what is most likely, then on election day when most likely becomes least likely, he washes his hands off. Thats just clearcut fraud. All Taleb is saying is that you simply don’t have ergodicity here so you don’t have a probability. You just have a [0..1] fraction that doesn’t mean much because it doesn’t converge to a limit as information approaches infinity. So if you had skin in the game and purchased an option based on the projected outcome you would lose big time.
- edna314 7y agoNaively I would have thought that there is another definition which goes along the lines that a process is ergodic if it looses memory of its starting conditions. But, maybe I’m mixing up things here.
- gowld 7y agoThat's essentially equivalent except in pathological synthetic cases, as time goes to infinity. Since any sample is representative of the whole, samples that don't include the intial conditions are as representative of the whole as samples that do include the initial conditions.
- hgibbs 7y agoWell, trivially, a dynamical system with that switches between two states is ergodic. We always know what the past state of the system was given the present. No, ergodicity is more about visiting the each part of the phase space with some positive frequency i.e. on a long trajectory you see everything that can happen, and not too infrequently.
- fasteo 7y agoThere is a recent paper[1] and an epic twitter thread[2] well worth the reading [1] https://arxiv.org/pdf/1906.04652.pdf https://arxiv.org/pdf/1906.04652.pdf [2] https://twitter.com/hulme_oliver/status/1139148255969906689 https://twitter.com/hulme_oliver/status/1139148255969906689
- mturmon 7y agoThe twitter thread gives a nice example of a failure of ergodicity that I was not aware of: "Now consider a gamble with multiplicative dynamics. Win 50% of your current wealth for heads, lose 40% of your current wealth for tails. Changes in wealth now are non-ergodic, so calculating the expectation value is not informative of the time average growth rate of wealth. This exact gamble has a positive expectation value, but it has a negative time average growth rate. We call it the Peters coin game." Clearly the expectation of return at a fixed time is > 0, but they give an example time series with obvious negative drift.
- dr_dshiv 7y agoI'm confused. My takeaway was that dice rolls and coin tosses are ergodic because flipping one coin 20 times or flipping 20 coins is meaningfully equivalent. Whereas following an individual person for 1000 days is different from following 1000 people for 1 day. Is that not the right heuristic? Also -- why is it called ergodicity? I keep thinking it has something to do with work or thermodynamics, but I'm missing it...
- cs702 7y agoIn addition to the paper and Twitter thread mentioned by fasteo elsewhere on this thread[a], this video of a lecture by the author is well-worth watching too: https://www.youtube.com/watch?v=f1vXAHGIpfc https://www.youtube.com/watch?v=f1vXAHGIpfc Among other things, in this lecture the author shows, step by step, in non-technical language, an example with coin tosses in which changes in wealth are non-ergodic (in the sense he explains), such that the expected value of winnings over time is not informative of the time average growth rate of wealth. [a] https://news.ycombinator.com/item?id=20285638 https://news.ycombinator.com/item?id=20285638
- ska 7y agoGetting a good feel for both stationarity and ergodicity in systems is worthwhile; assuming them is a common way to “intuitively” infer incorrect behaviour.
- Liquix 7y agoGreat read. Eat your heart out, Betteridge's Law!
- claudiawerner 7y agoThe author writes, >Paul Samuelson once famously claimed that the “ergodic hypothesis” is essential for advancing economics from the realm of history to the realm of science. It's curious that science and history are pitted against each other; not only is science notoriously difficult to define, but there are several historical approaches to scientific objects (e.g archaeology for the object of human history). This also seems to involve a logical model in which the concepts stay static, but some popular models outside of mainstream economics are used because the authors argue that an approach divorced from real development leads to results that don't apply to real situations (e.g. the dialectical logical method, which is both syntactic and semantic. Unfortunately the sort of logic Samuelson applies has been misapplied (in neo-Ricardian lenses) to thinkers which seem to hold strictly temporal (rather than equilibrium) interpretations of economy.
- selimthegrim 7y agoErgodic and chaotic are not the same either (the difference being in long time diverging behavior of two trajectories that start very close together); many lazy physics science writers and occasionally physicists conflate them.