Please consider making a video on the application of ergodicity to portfolio theory in finance! Many people who already understand the Kelly Criterion would be interested in the broader applications of ergodicity
According to chatgpt: The reference to Feynman’s lectures likely points to Richard Feynman’s “Lectures on Physics,” which cover a wide range of topics in physics, including statistical mechanics and quantum mechanics, where such stochastic properties are discussed. The specific mention of an induction argument suggests the explanation involves building understanding step-by-step, a common pedagogical method in Feynman’s work.
This takes me to the limit of my calculus recall (50 years ago) but I've been running Monte Carlo simulations in Excel which helps me understand. It is incredibly fascinating, thank you.
Great video. Expanding on the link with Kelly criterion would be fantastic (specifically the influence of random "lucky" trajectories in Kelly weightings.)
OHMYGOD, my brain is blown! (1.5+0.6)/2 = 1.05 increases the 'collective' wealth, but 1.5*0.6 = 0.9 decreases the individual's wealth! Question for Ole Peters: You reconcile these two facts saying there are some infiinitely rare trajectories with very high growth factors. But how do you even know that these trajectories exist if they are infinitely rare?
To put the question another way: how do you know that a fair coin can come up heads 1 million times in a row if it's infinitely rare in practice? It's part of the definition of "fair," that's all. If we take the limit as the number of games goes to infinity, we end up averaging over all circumstances that "can" happen.
I am wondering how is this related to the concept of stationarity in time series analysis, where stationarity is stated as ensemble statistics, but measure as time statistics.
Professor: "The probability that your trajectory in the time perspective of this bet will go up is 1/(10^1500). Me, an intellectual: So you're saying there's a chance...
Is this the reason for the difference between the arithmetic average and the geometric average? For example growth and contraction of both 50% for six periods. Starting at 100, the sequence is 150,75,113,56,84,42. The arithmetic average is 0%, the geometric growth is negative.
Great video, very well explained. Could it be possible if you could do some videos where you show if a known stochastic process is ergodic or not? Specifically, could you do a video on bitcoin and fiat currency? Thanks.
Thanks for the material! I don't know how to reconcile Mandelbrot's non-gaussian increments with EE. Can you help me? Is it a serious contradiction or irrelevant to main conclusions?
The two are more or less unrelated. In the video I use two kinds of increments: binomial ones, i.e. two discrete possible values, and Gaussian ones. Ergodicity is broken in the same way, as you can see somewhere towards the end. The origin of ergodicity breaking is deeper than the type of distribution of increments. You can use, say, power-law distributed increments if you like. Depending on how fat the tails are, this will introduce additional problems, but the ergodicity breaking is not affected by it. There is a longer story here. For example, if you're drawing increments from a stationary distribution whose variance does not exist because of fat tails, and you measure the finite-sample variance, you will see it systematically increase with the sample size. This can look very similar to drawing from a non-stationary (non-ergodic) distribution - say from a Gaussian whose width increases with time. In other words, sometimes the same phenomenon can be reasonably modeled with a fat-tailed distribution or a non-ergodic process.
Hmm, this makes me a little concerned about expected value utilitarianism as operated by people who believe in the multiverse hypothesis. Call me old-fasioned, but I rather don't like the prospect of being in a decaying world for the benefit of nearly infinitesimally few unfalsifiable but extremely counterfactually lucky folks in a different universe. My goodness, I hope I don't seem greedy to my future betters. Then again, I'm not sure I'll have them, what with AGI likely within a few decades. My what a conundrum.
I’m truly puzzled by the statement that the winning trajectories are extremely rare while the losing trajectories are many. In the game since your winning and losing odds are equal, why aren’t the two trajectories equally likely to occur, and then each grows exponentially at 5%? Why do we (nature) tend to favor losing trajectories?
It's because they're multiplied rather than added I think. If you win more, you have a larger chunk to lose when you lose. I don't fully understand myself though.
Well, what are "the two" trajectories? With k tosses, there are 2^k possible trajectories. The vast majority of them will have a roughly equal number of heads and tails. Call that number 'n'. Then you will have 0.6^n * 1.5^n = 0.9^n ~= 0 by the end. Only the ones with significantly more heads will be winning, and those become increasingly rare (for the same reason that the odds of getting 60% heads goes to zero as the number of tosses of a fair coin goes to infinity).
For the game that you suggest i.e 50/40 with 50/50 chance you argue that the optimal bet is 0.25... Isn't fraction supposed to be 0.10 as suggested by Kelly criterion?
@@William.-. wait what? What are the 2 formulas? In my head kelly fraction is the fraction that maximises geometric return, regardless of how you write it using log or powers u get the same result.
@@denisryabich2601 Yea you are correct I think. What i meant by two different Kelly formula is that 0.10 uses the formula where you lose 100% of the bank roll, it's also the "popular" formula that people referenced. Whereas the 0.25 formula which is referenced on Wikipedia uses volality drag, the higher the lose, the lower you should bet in case you get it wrong if you do it more than once.
Is it a paradox or simply that the ensemble distribution in any period has a very long tail? If you take the median or geometric mean of the ensemble at each time point, it seems to behave much like the time average.
You're taking a sample mean of the ensemble , which is different to the expectation of ensemble. Its mentioned in the video that the E[] term implicitly assumes an infinite number of ensembles, so it's no paradox that there are very lucky outcomes in that infinity.
8:08: "Imagine an economy where income behaves in this way. GDP would grow but practically everyones income would decline." "If such conditions persists for a long time, what happens to democracy?"
@8:30 you are a little "dopey" here, in good ol' Feynman lingo. "What happens to democracy?" Well, if you actually _had_ democracy to begin with, plus an understanding of state monetary systems (tax driven, fiat), then there would be no random exchange dynamics (unless you choose it to be so), nor Pareto dynamics (unless again, by choice, or for chosen unregulated markets). You could instead simply vote to distribute wealth fairly (real and nominal), no one "trajectory" (family or person) would suffer. I know it is a bit passé, but it is still a decent sort of starting principle if we append a few realist words: _from each according to their ability, to each according to their needs, up to available real resource constraints._
It's very important to note that stock returns aren't gaussian. Therefore, using the average and standard deviation for this model when applied to stock markets, is flawed. As Ole says: systems in nature are gaussian. This is because there are physical limitations/boundaries towards the extremes of the distribution. You'll never see a person that's 4 meters tall, which means human height is gaussian. When looking at finance and economics, there aren't many boundaries. It is more likely to lose 90% of your portfolio, than a gaussian would predict. Wealth of individuals are also not subject to boundaries and follow a power law. Be very careful with the model Ole describes and only apply it to things that have a natural boundary, so almost nothing in finance or economics.
Am a bachelor student in physics working with stochastic diff. eq. and found your video very helpful!
Thanks!
Please consider making a video on the application of ergodicity to portfolio theory in finance! Many people who already understand the Kelly Criterion would be interested in the broader applications of ergodicity
Link to Feynman lecture referenced in the video?
According to chatgpt: The reference to Feynman’s lectures likely points to Richard Feynman’s “Lectures on Physics,” which cover a wide range of topics in physics, including statistical mechanics and quantum mechanics, where such stochastic properties are discussed. The specific mention of an induction argument suggests the explanation involves building understanding step-by-step, a common pedagogical method in Feynman’s work.
Come for a revolution in economics, stay for the flowers and the chirping of birds.
This is amazing. Please publish more videos. Quite inspiring truly.
This takes me to the limit of my calculus recall (50 years ago) but I've been running Monte Carlo simulations in Excel which helps me understand. It is incredibly fascinating, thank you.
This is an excellent explanation, thank you Ole
Awesome ... I am looking forward to the other videos!
Great video. Expanding on the link with Kelly criterion would be fantastic (specifically the influence of random "lucky" trajectories in Kelly weightings.)
Thanks for sharing, Ole!
Lovely video Ole. Keep'em coming :)
Agreed!
OHMYGOD, my brain is blown!
(1.5+0.6)/2 = 1.05 increases the 'collective' wealth, but 1.5*0.6 = 0.9 decreases the individual's wealth!
Question for Ole Peters: You reconcile these two facts saying there are some infiinitely rare trajectories with very high growth factors. But how do you even know that these trajectories exist if they are infinitely rare?
To put the question another way: how do you know that a fair coin can come up heads 1 million times in a row if it's infinitely rare in practice? It's part of the definition of "fair," that's all. If we take the limit as the number of games goes to infinity, we end up averaging over all circumstances that "can" happen.
I am wondering how is this related to the concept of stationarity in time series analysis, where stationarity is stated as ensemble statistics, but measure as time statistics.
Thanks Ole
Thanks Ole!
Professor: "The probability that your trajectory in the time perspective of this bet will go up is 1/(10^1500).
Me, an intellectual: So you're saying there's a chance...
Errhmmmm... N-sigma ? What is N?
Great Video. Thanks
Is this the reason for the difference between the arithmetic average and the geometric average? For example growth and contraction of both 50% for six periods. Starting at 100, the sequence is 150,75,113,56,84,42. The arithmetic average is 0%, the geometric growth is negative.
This is excellent. Thank you!!
thank you, this is fascinating stuff
Great video, very well explained. Could it be possible if you could do some videos where you show if a known stochastic process is ergodic or not? Specifically, could you do a video on bitcoin and fiat currency? Thanks.
Thanks for the material! I don't know how to reconcile Mandelbrot's non-gaussian increments with EE. Can you help me? Is it a serious contradiction or irrelevant to main conclusions?
The two are more or less unrelated. In the video I use two kinds of increments: binomial ones, i.e. two discrete possible values, and Gaussian ones. Ergodicity is broken in the same way, as you can see somewhere towards the end. The origin of ergodicity breaking is deeper than the type of distribution of increments. You can use, say, power-law distributed increments if you like. Depending on how fat the tails are, this will introduce additional problems, but the ergodicity breaking is not affected by it.
There is a longer story here. For example, if you're drawing increments from a stationary distribution whose variance does not exist because of fat tails, and you measure the finite-sample variance, you will see it systematically increase with the sample size. This can look very similar to drawing from a non-stationary (non-ergodic) distribution - say from a Gaussian whose width increases with time. In other words, sometimes the same phenomenon can be reasonably modeled with a fat-tailed distribution or a non-ergodic process.
Hmm, this makes me a little concerned about expected value utilitarianism as operated by people who believe in the multiverse hypothesis.
Call me old-fasioned, but I rather don't like the prospect of being in a decaying world for the benefit of nearly infinitesimally few unfalsifiable but extremely counterfactually lucky folks in a different universe.
My goodness, I hope I don't seem greedy to my future betters.
Then again, I'm not sure I'll have them, what with AGI likely within a few decades.
My what a conundrum.
Great Video
I’m truly puzzled by the statement that the winning trajectories are extremely rare while the losing trajectories are many. In the game since your winning and losing odds are equal, why aren’t the two trajectories equally likely to occur, and then each grows exponentially at 5%? Why do we (nature) tend to favor losing trajectories?
It's because they're multiplied rather than added I think. If you win more, you have a larger chunk to lose when you lose. I don't fully understand myself though.
Well, what are "the two" trajectories? With k tosses, there are 2^k possible trajectories. The vast majority of them will have a roughly equal number of heads and tails. Call that number 'n'. Then you will have 0.6^n * 1.5^n = 0.9^n ~= 0 by the end. Only the ones with significantly more heads will be winning, and those become increasingly rare (for the same reason that the odds of getting 60% heads goes to zero as the number of tosses of a fair coin goes to infinity).
For the game that you suggest i.e 50/40 with 50/50 chance you argue that the optimal bet is 0.25... Isn't fraction supposed to be 0.10 as suggested by Kelly criterion?
Two different Kelly criterion formula
@@William.-. wait what? What are the 2 formulas? In my head kelly fraction is the fraction that maximises geometric return, regardless of how you write it using log or powers u get the same result.
@@denisryabich2601 Yea you are correct I think. What i meant by two different Kelly formula is that 0.10 uses the formula where you lose 100% of the bank roll, it's also the "popular" formula that people referenced.
Whereas the 0.25 formula which is referenced on Wikipedia uses volality drag, the higher the lose, the lower you should bet in case you get it wrong if you do it more than once.
@@denisryabich2601 also how did you get 0.10 or 10%?
Excellent! Thanks! 🙂 👏
Is it a paradox or simply that the ensemble distribution in any period has a very long tail? If you take the median or geometric mean of the ensemble at each time point, it seems to behave much like the time average.
You're taking a sample mean of the ensemble , which is different to the expectation of ensemble. Its mentioned in the video that the E[] term implicitly assumes an infinite number of ensembles, so it's no paradox that there are very lucky outcomes in that infinity.
8:08: "Imagine an economy where income behaves in this way. GDP would grow but practically everyones income would decline." "If such conditions persists for a long time, what happens to democracy?"
@8:30 you are a little "dopey" here, in good ol' Feynman lingo. "What happens to democracy?" Well, if you actually _had_ democracy to begin with, plus an understanding of state monetary systems (tax driven, fiat), then there would be no random exchange dynamics (unless you choose it to be so), nor Pareto dynamics (unless again, by choice, or for chosen unregulated markets). You could instead simply vote to distribute wealth fairly (real and nominal), no one "trajectory" (family or person) would suffer. I know it is a bit passé, but it is still a decent sort of starting principle if we append a few realist words: _from each according to their ability, to each according to their needs, up to available real resource constraints._
Do you need money to buy a comb? I could Venmo it to you
It's very important to note that stock returns aren't gaussian. Therefore, using the average and standard deviation for this model when applied to stock markets, is flawed.
As Ole says: systems in nature are gaussian. This is because there are physical limitations/boundaries towards the extremes of the distribution. You'll never see a person that's 4 meters tall, which means human height is gaussian.
When looking at finance and economics, there aren't many boundaries. It is more likely to lose 90% of your portfolio, than a gaussian would predict. Wealth of individuals are also not subject to boundaries and follow a power law.
Be very careful with the model Ole describes and only apply it to things that have a natural boundary, so almost nothing in finance or economics.