<p>In physics, an observable (i.e. quantity) is said to fulfil the ergodic hypothesis if both its ensemble average (phase average) and its time average coincide, i.e. yield the same result.</p>
<p>In his paper, Peters investigates the exponential growth rate in a given stochastic process-related setting. He uses an abstraction of the ergodic hypothesis to be probed by observables of these processes. He finds that for the studied dynamics the exponential growth rate fulfils the ergodic hypothesis, i.e. the expectation value of the exponential growth rate and its time average yield the same result. For this specific process, the two averages are indistinguishable, which is why we can use the one quantity to inform the other. In particular, we can use the expectation value to tell us what happens over time (or &quot;in the long run&quot;).</p>
<p>In contrast, additive wealth changes in the given process-related setting do not fulfil the ergodic hypothesis. That is, the two averages of said quantity do not yield the same result and we cannot simply replace the one average of the quantity with the other. In particular, we cannot use the expectation value to tell us what happens over time for a quantity (i.e. observable) that does not fulfil the ergodic hypothesis.</p>

CC BY-SA 4.0
2024-12-27T09:53:26.883Z
Ergodicity is a property of measure-preserving transformations: They are ergodic if there are no nontrivial invariant sets.
Michael Greinecker
CC BY-SA 4.0
2024-12-28T00:04:03.770Z
Certainly not in the context in which it is used here. I gave the proper definition in the given context in my first sentence. This definition stems from equilibrium statistical mechanics (cf. the Ergodicity Hypothesis).
NeverMind
CC BY-SA 4.0
2024-12-28T10:31:40.837Z
The mathematical definition in terms of measure-preserving operators comes from statistical physics, going back all the way to Poincare. And while Peters may not know the relevant math, I would assume Taleb does.
Michael Greinecker
CC BY-SA 4.0
2024-12-28T16:21:17.193Z
But what is your point? Peters simply uses Birkhoff’s ergodic theorem to test the Ergodic hypothesis for a given dynamic observable. There certainly is more formal math behind ergodic theory, but in the end what he cares about is the question of equality of ensemble average and time average, and whether we need to be cautious about which one to choose for a specific problem/perspective.
NeverMind
CC BY-SA 4.0
2024-12-28T18:41:45.643Z
Well, Birkhoff's ergodic theorem is not about your definition. So I guess you answered your question.
Michael Greinecker
CC BY-SA 4.0
2024-12-29T10:05:22.560Z
I edited my answer to strictly separate between the ergodic hypothesis and ergodic theory. Peters was trained as a theoretical physicist, so he should be familiar with Boltzmann's work about the ergodic hypothesis and such things as the Liouville equation. He is not necessarily familiar with ergodic theory as an abstract mathematical theory and part of general dynamics. Note that even with this distinction, Peters' usage of testing the ergodic hypothesis is only an abstraction as in his problems there is no phase space and physical ensemble defined.
NeverMind
CC BY-SA 4.0
2024-12-30T00:06:48.587Z
NeverMind. Just one more question. Do you have know of a nice and reasonably thorough explanation of ergodicity. My math is at the level of real analysis and measure theory but far from mathematician or physicist. Thanks.
mark leeds
CC BY-SA 4.0
2024-12-30T14:21:13.293Z
Maybe one practical way to think about it is in the form of the ergodic hypothesis from physics, which roughly states that an ergodic system in free evolution for a sufficiently long time will pass through all its microstates consistent with its general conditions (total energy). From this general notion important consequences can be derived, e.g. that the average time a system spends in a phase region is proportional to the measure of the region itself and that one and only one probability distribution exists that is invariant with regard to the transformations taking place on the system.
NeverMind
CC BY-SA 4.0
2024-12-30T14:35:31.047Z
In equilibrium statistical physics, the assumption of the ergodic hypothesis allows us to describe the long time-behavior of macroscopic observables of a system in terms of its ensemble average, i.e. an average over infinite virtual copies of the system, each of which represents a possible microstate that the real system might be in. By that, we effectively eliminate time from the macroscopic equations of motion and connect a set theoretical quantity, the ensemble or phase average of a function, with a quantity defined on the system, that is, the infinite time average of a physical quantity.
NeverMind
CC BY-SA 4.0
2024-12-30T17:20:12.697Z
Thanks Never Mind. Your thorough comments are appreciated. The terminolgy is somewhat difficult for me because background is statistics but there is quite a bit wisdom in this thread that I can use to get a better handle on it. Thanks again.
mark leeds
CC BY-SA 4.0
2024-12-26T23:57:16.027Z
2024-12-29T11:28:39.403Z
2024-12-29T11:28:39.403Z
NeverMind
<p>You may or may not be aware that there's a <a href="https://simple.wikipedia.org/wiki/Main_Page" rel="nofollow noreferrer">Simple English Wikipedia</a>.</p>
<p>It's very helpful for those of us who know English, but are unable to parse complex sentences (for whatever reason).</p>
<p>I'm very interested in Taleb's concept of Ergodicity (Taleb may have gotten the idea from Mandelbrot).</p>
<p><a href="https://ergodicityeconomics.com/author/olebpeters/" rel="nofollow noreferrer">Ole Peters</a> has probably been most responsible for fleshing out the concept in the domain of Economics.</p>
<p>I'm very intrigued by the concept and am curious about ways to apply it in my life to improve my life. You may be too.</p>
<p>I know some math, but am often unable to interpret complex mathematical concepts.</p>
<p>I've posted a screenshot below with three sentences/phrases underlined. The screenshot is from <a href="https://ergodicityeconomics.com/2018/02/16/the-trouble-with-bernoulli-1738/" rel="nofollow noreferrer"><strong>this article</strong></a> by Ole. I'd appreciate if you could provide a simple breakdown of what one or more of these sentences are trying to say. Thanks in advance.</p>
<p><a href="https://i.sstatic.net/LEzdt.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/LEzdt.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-11-10T20:08:28.933Z
I would not pay too much attention to what Ole Peters is saying... See the [consensus](https://economics.stackexchange.com/questions/47097) on his contributions.
Richard Hardy
CC BY-SA 4.0
2023-11-11T02:17:39.957Z
Truth be told: when I got to the line "Economics textbooks and papers miss this point" I stopped reading. Because, unfortunately, economics attracts a lot of I'm-right-and-conventional-econ-is-wrong people. My advice: anyone promoting a heterodox theory should briefly explain their changes to conventional econ and how it improves the result.
Daniel
CC BY-SA 4.0
2024-12-28T01:05:12.853Z
@RichardHardy Sorry to learn that you think "consensus" drives Science forward. It didn't when Semmelweis was institutionalized, and it didn't when Wegener was resoundingly mocked for suggesting that India was once part of Africa, and drifted away. Enjoy being in your bubble of "consensus". Ole Peters' ideas have transformed my understanding of Economics, and I will continue to enjoy them, thank you very much. The last thing I need is the "consensus opinion of Economists". As Taleb would say, show me your P&L.
thanks_in_advance
CC BY-SA 4.0
2024-12-28T02:55:53.757Z
@Daniel "Conventional Economics" is regularly proven wrong (in its predictions). Atrociously so. Behavioral Economics has led to a revision in Economics textbooks which is far from complete: [Daniel Kahneman](https://www.nobelprize.org/prizes/economic-sciences/2002/kahneman/lecture/) , [Homo Economicus](https://en.wikipedia.org/wiki/Homo_economicus) . The entire paper is about how Ole's heterdox theory changes conventional econ and improves the result. That's what the paper is about.
thanks_in_advance
CC BY-SA 4.0
2024-12-28T20:23:49.890Z
You have learned an incorrect thing about what I think (you invented an opinion that I do not hold), but I am relieved you are sorry about it.
Richard Hardy
<expected-utility><soft-question>
Bernoulli, Ergodicity, Ole Peters
thanks_in_advance
# Bernoulli, Ergodicity, Ole Peters

(Asked by: username_0 on 2023-11-10)

You may or may not be aware that there's a [Simple English Wikipedia](https://simple.wikipedia.org/wiki/Main_Page).

It's very helpful for those of us who know English, but are unable to parse complex sentences (for whatever reason).

I'm very interested in Taleb's concept of Ergodicity (Taleb may have gotten the idea from Mandelbrot).

[Ole Peters](https://ergodicityeconomics.com/author/olebpeters/) has probably been most responsible for fleshing out the concept in the domain of Economics.

I'm very intrigued by the concept and am curious about ways to apply it in my life to improve my life. You may be too.

I know some math, but am often unable to interpret complex mathematical concepts.

I've posted a screenshot below with three sentences/phrases underlined. The screenshot is from [**this article**](https://ergodicityeconomics.com/2018/02/16/the-trouble-with-bernoulli-1738/) by Ole. I'd appreciate if you could provide a simple breakdown of what one or more of these sentences are trying to say. Thanks in advance.

[![enter image description here](https://i.sstatic.net/LEzdt.jpg)](https://i.sstatic.net/LEzdt.jpg)

**Tags:** expected-utility, soft-question
> **Comments:**
> **username_1**: I would not pay too much attention to what Ole Peters is saying... See the [consensus](https://economics.stackexchange.com/questions/47097) on his contributions.  
> **username_2**: Truth be told: when I got to the line "Economics textbooks and papers miss this point" I stopped reading. Because, unfortunately, economics attracts a lot of I'm-right-and-conventional-econ-is-wrong people. My advice: anyone promoting a heterodox theory should briefly explain their changes to conventional econ and how it improves the result.  
> **username_0**: @RichardHardy Sorry to learn that you think "consensus" drives Science forward. It didn't when Semmelweis was institutionalized, and it didn't when Wegener was resoundingly mocked for suggesting that India was once part of Africa, and drifted away. Enjoy being in your bubble of "consensus". Ole Peters' ideas have transformed my understanding of Economics, and I will continue to enjoy them, thank you very much. The last thing I need is the "consensus opinion of Economists". As Taleb would say, show me your P&L.  
> **username_0**: @Daniel "Conventional Economics" is regularly proven wrong (in its predictions). Atrociously so. Behavioral Economics has led to a revision in Economics textbooks which is far from complete: [Daniel Kahneman](https://www.nobelprize.org/prizes/economic-sciences/2002/kahneman/lecture/) , [Homo Economicus](https://en.wikipedia.org/wiki/Homo_economicus) . The entire paper is about how Ole's heterdox theory changes conventional econ and improves the result. That's what the paper is about.  
> **username_1**: You have learned an incorrect thing about what I think (you invented an opinion that I do not hold), but I am relieved you are sorry about it.  

---

## 1 answers

### Answer by username_3 on 2024-12-26. Score: 1

In physics, an observable (i.e. quantity) is said to fulfil the ergodic hypothesis if both its ensemble average (phase average) and its time average coincide, i.e. yield the same result.

In his paper, Peters investigates the exponential growth rate in a given stochastic process-related setting. He uses an abstraction of the ergodic hypothesis to be probed by observables of these processes. He finds that for the studied dynamics the exponential growth rate fulfils the ergodic hypothesis, i.e. the expectation value of the exponential growth rate and its time average yield the same result. For this specific process, the two averages are indistinguishable, which is why we can use the one quantity to inform the other. In particular, we can use the expectation value to tell us what happens over time (or "in the long run").

In contrast, additive wealth changes in the given process-related setting do not fulfil the ergodic hypothesis. That is, the two averages of said quantity do not yield the same result and we cannot simply replace the one average of the quantity with the other. In particular, we cannot use the expectation value to tell us what happens over time for a quantity (i.e. observable) that does not fulfil the ergodic hypothesis.

> **Comments:**
> **username_4**: Ergodicity is a property of measure-preserving transformations: They are ergodic if there are no nontrivial invariant sets.  
> **username_3**: Certainly not in the context in which it is used here. I gave the proper definition in the given context in my first sentence. This definition stems from equilibrium statistical mechanics (cf. the Ergodicity Hypothesis).  
> **username_4**: The mathematical definition in terms of measure-preserving operators comes from statistical physics, going back all the way to Poincare. And while Peters may not know the relevant math, I would assume Taleb does.  
> **username_3**: But what is your point? Peters simply uses Birkhoff’s ergodic theorem to test the Ergodic hypothesis for a given dynamic observable. There certainly is more formal math behind ergodic theory, but in the end what he cares about is the question of equality of ensemble average and time average, and whether we need to be cautious about which one to choose for a specific problem/perspective.  
> **username_4**: Well, Birkhoff's ergodic theorem is not about your definition. So I guess you answered your question.  
> **username_3**: I edited my answer to strictly separate between the ergodic hypothesis and ergodic theory. Peters was trained as a theoretical physicist, so he should be familiar with Boltzmann's work about the ergodic hypothesis and such things as the Liouville equation. He is not necessarily familiar with ergodic theory as an abstract mathematical theory and part of general dynamics. Note that even with this distinction, Peters' usage of testing the ergodic hypothesis is only an abstraction as in his problems there is no phase space and physical ensemble defined.  
> **username_5**: NeverMind. Just one more question. Do you have know of a nice and reasonably thorough explanation of ergodicity. My math is at the level of real analysis and measure theory but far from mathematician or physicist. Thanks.  
> **username_3**: Maybe one practical way to think about it is in the form of the ergodic hypothesis from physics, which roughly states that an ergodic system in free evolution for a sufficiently long time will pass through all its microstates consistent with its general conditions (total energy). From this general notion important consequences can be derived, e.g. that the average time a system spends in a phase region is proportional to the measure of the region itself and that one and only one probability distribution exists that is invariant with regard to the transformations taking place on the system.  
> **username_3**: In equilibrium statistical physics, the assumption of the ergodic hypothesis allows us to describe the long time-behavior of macroscopic observables of a system in terms of its ensemble average, i.e. an average over infinite virtual copies of the system, each of which represents a possible microstate that the real system might be in. By that, we effectively eliminate time from the macroscopic equations of motion and connect a set theoretical quantity, the ensemble or phase average of a function, with a quantity defined on the system, that is, the infinite time average of a physical quantity.  
> **username_5**: Thanks Never Mind. Your thorough comments are appreciated. The terminolgy is somewhat difficult for me because background is statistics but there is quite a bit wisdom in this thread that I can use to get a better handle on it. Thanks again.  

---

