A random process is ergodic if all of its statistics can be determined from a sample function of the process. That is, the ensemble averages equal the corresponding time averages with probability one.
Role of Ergodicity in Human Inference
A newspaper has previously printed some inaccurate information, therefore, the newspaper is going to publish inaccurate information in the future. Fair; ensemble of published articles is more or less ergodic.
More crimes are committed by black persons than by white persons, therefore each individual black person is not to be trusted? The ensemble of black people is not at all ergodic!
Ergodic ≠ Stationary
Although all ergodic processes are stationary (unconditional joint PDF doesn’t change in time; consequently parameters such as \(\mu, \sigma\) don’t change over time), they are not equivalent:
- Say we have coin A with \( \mathbb{P}\{H\} =.5 \) and coin B with \( \mathbb{P}\{H\} = .25 \)
- We pick either of the coins with \( p = .5\)
- We toss the coin that we picked repeatedly, noting \(H_1, H_2, H_n\)
- Although the process is stationary, the time average \(H_1 + … + H_n \) either converges to \(\frac{1}{2}\) or \(\frac{1}{4} \) with equal probability.
- The time average depends on the coin you choose, while the probabilistic average \(\left(\frac{3}{8} \right) \) is calculated for the whole system.
An Ergodic System (In a Sense)
Say I gave \(n\) people a die each, had them roll their die once. \( \frac{X_1 + … + X_n}{n} \) is a finite-sample average which approaches the ensemble average as \(n \to \infty \).
Say I rolled a die \(t\) times and calculate \(\frac{X_1 + … + X_t}{t}\). This finite-time average approaches the time average as \(t \to \infty \).
One implication of ergodicity is that ensemble averages will be the same as time averages. In the first case, as \(n \to \infty \) the randomness is removed from the system. In the second case, as \(t \to \infty \) the randomness is removed. But both methods give the same answer, within errors.
Notes on Ergodicity
In this sense, rolling dice is an ergodic system. But if we bet on the results of rolling a die, wealth does not follow an ergodic process under typical betting rules. If I go bankrupt, I’ll stay bankrupt. So the time average of my wealth will approach zero as time passes, even though the ensemble average of my wealth may increase.
Stationarity is required for ergodicity, for there can be no growth in an ergodic system. Ergodic systems are zero-sum games. No branching occurs in an ergodic system, no decision has any consequences because sooner or later we’ll end up in the same situation again and can reconsider.
The key is that most systems of interest to us, including finance, are non-ergodic.
So no free lunch. From a SWE perspective, value comes from solving people’s problems. As long as human systems’s are non-ergodic, then big tech is still poised to develop moats – small tech doesn’t have as many resources to approximate \(n \to \infty\) or \(t \to \infty\).
References
- What is Ergodicity? Lars P. Syll. larspsyll.wordpress.com . Nov 23, 2016.
- Assessing the Race–Crime and Ethnicity–Crime Relationship in a Sample of Serious Adolescent Delinquents. Alex R Piquero; Robert W Brame. Crime & Delinquency, Vol 54. #3. doi.org . pmc.ncbi.nlm.nih.gov . 2008. Accessed Dec 25, 2024.
larspsyll.wordpress.com
This statistic is sometimes used in a dog whistle sense to justify anti-black racism. Is it a “differential involvement hypothesis” where Blacks simply commit more crime into adulthood? Or is it a “differential criminal justice system selection hypothesis” where differential police presence, patrolling, profiling, and judicial discrimination leads to more Blacks being arrested, convicted and incarcerated? However, when filtered to serious offender populations, there is little evidence to support the relationship between race and criminal involvement.