What Is Ergodicity?

Individual behavior and ensembles

Why are election polls often inaccurate? Why is racism wrong? Why are your assumptions often mistaken? The answers to all these questions and to many others have a lot to do with the non-ergodicity of human ensembles. Many scientists agree that ergodicity is one of the most important concepts in statistics. So, what is it?

Ergodicity is usually described in terms of objective properties of an ensemble of objects, and the discussion often gets lost in mathematical subtleties and thus it is often difficult to understand. Nonetheless, I will describe it in bayesian, subjectivist terms; hopefully this will make the concept very accessible.

Suppose you are concerned with determining what the most visited parks in a city are. One idea is to take a momentary snapshot: to see how many people are this moment in park A, how many are in park B and so on. Another idea is to look at one individual (or few of them) and to follow him for a certain period of time, e.g. a year. Then, you observe how often the individual is going to park A, how often he is going to park B and so on.

Thus, you obtain two different results: one statistical analysis over the entire ensemble of people at a certain moment in time, and one statistical analysis for one person over a certain period of time. The first one may not be representative for a longer period of time, while the second one may not be representative for all the people.

The idea is that an ensemble is ergodic if the two types of statistics give the same result. Many ensembles, like the human populations, are not ergodic.

The importance of ergodicity becomes manifest when you think about how we all infer various things, how we draw some conclusion about something while having information about something else. For example, one goes once to a restaurant and likes the fish and next time he goes to the same restaurant and orders chicken, confident that the chicken will be good. Why is he confident? Or one observes that a newspaper has printed some inaccurate information at one point in time and infers that the newspaper is going to publish inaccurate information in the future. Why are these inferences ok, while others such as "more crimes are committed by black persons than by white persons, therefore each individual black person is not to be trusted" are not ok?

The answer is that the ensemble of articles published in a newspaper is more or less ergodic, while the ensemble of black people is not at all ergodic. If one searches how many mistakes appear in an entire newspaper in one issue, and then searches how many mistakes one news editor does over time, one finds the two results almost identical (not exactly, but nonetheless approximately equal). However, if one takes the number of crimes committed by black people in a certain day divided by the total number of black people, and then follows one random-picked black individual over his life, one would not find that, e.g. each month, this individual commits crimes at the same rate as the crime rate determined over the entire ensemble. Thus, one cannot use ensemble statistics to properly infer what is and what is not probable that a certain individual will do.

Or take an even clearer example: In an election each party gets some percentage of votes, party A gets a%, party B gets b% and so on. However, this does not mean that over the course of their lives each individual votes with party A in a% of elections, with B in b% of elections and so on.

These were examples of why, in some cases - the non-ergodic cases, one cannot use ensemble statistics to infer something about a particular individual. There is also a complementary problem, faced by the scientists doing opinion polls. They gather data from a very small number of individuals and try to infer the characteristics of the entire ensemble. In order to do this as accurately as possible they don't simply pick the individuals at random; they partition the human ensemble on the basis of some criteria (such as age or income) and afterwards they randomly pick individuals inside each partition being careful that each partition is being represented. It is worth noting that the so-called margin of error of the opinion polls is not really a margin of error. This margin of error is computed assuming that the human ensemble (or more precisely, the partitions they establish) is (are) ergodic. But in reality they are not.

A similar problem is faced by scientists in general when they are trying to infer some general statement from various particular experiments. When is a generalization correct and when it isn't? The answer concerns ergodicity. If the generalization is done towards an ergodic ensemble, than it has a good chance of being correct

Milton Friedman's 20 best quotes

Yesterday(2015.8.1) would have been the 103^rd birthday of Milton Friedman, who was one of the most brilliant economists of the last century. In honor of Friedman, here are his 20 best quotes.


20) “A society that puts equality before freedom will get neither. A society that puts freedom before equality will get a high degree of both.”
19) “Because we live in a largely free society, we tend to forget how limited is the span of time and the part of the globe for which there has ever been anything like political freedom: the typical state of mankind is tyranny, servitude, and misery. The nineteenth century and early twentieth century in the Western world stand out as striking exceptions to the general trend of historical development. Political freedom in this instance clearly came along with the free market and the development of capitalist institutions. So also did political freedom in the golden age of Greece and in the early days of the Roman era.”
18) “It is one thing to have free immigration to jobs. It is another thing to have free immigration to welfare. And you cannot have both. If you have a welfare state, if you have a state in which every resident is promised a certain minimal level of income, or a minimum level of subsistence, regardless of whether he works or not, produces it or not. Then it really is an impossible thing.”
17) “So that the record of history is absolutely crystal clear. That there is no alternative way, so far discovered, of improving the lot of the ordinary people that can hold a candle to the productive activities that are unleashed by a free enterprise system.”
16) "When everybody owns something, nobody owns it, and nobody has a direct interest in maintaining or improving its condition. That is why buildings in the Soviet Union - like public housing in the United States - look decrepit within a year or two of their construction…"
15) "The great danger to the consumer is the monopoly - whether private or governmental. His most effective protection is free competition at home and free trade throughout the world. The consumer is protected from being exploited by one seller by the existence of another seller from whom he can buy and who is eager to sell to him. Alternative sources of supply protect the consumer far more effectively than all the Ralph Naders of the world."
14) "Two major arguments are offered for introducing socialized medicine in the United States: first, that medical costs are beyond the means of most Americans; second that socialization will somehow reduce costs. The second can be dismissed out of hand -- at least until someone can find some example of an activity that is conducted more economically by the government than private enterprise. As to the first, the people of the country must pay the costs one way or the other; the only question is whether they pay them directly on their own behalf, or indirectly through the mediation of government bureaucrats who will subtract a substantial slice for their own salaries and expenses."
13) "Nothing is so permanent as a temporary government program."
12) "The supporters of tariffs treat it as self-evident that the creation of jobs is a desirable end, in and of itself, regardless of what the persons employed do. That is clearly wrong. If all we want are jobs, we can create any number - for example, have people dig holes and then fill them up again, or perform other useless tasks. Work is sometimes its own reward. Mostly, however, it is the price we pay to get the things we want. Our real objective is not just jobs but productive jobs - jobs that will mean more goods and services to consume."
11) "I am in favor of cutting taxes under any circumstances and for any excuse, for any reason, whenever it's possible."
10) "There is all the difference in the world, however, between two kinds of assistance through government that seem superficially similar: first, 90 percent of us agreeing to impose taxes on ourselves in order to help the bottom 10 percent, and second, 80 percent voting to impose taxes on the top 10 percent to help the bottom 10 percent - William Graham Sumner's famous example of B and C decided what D shall do for A. The first may be wise or unwise, an effective or ineffective way to help the disadvantaged - but it is consistent with belief in both equality of opportunity and liberty. The second seeks equality of outcome and is entirely antithetical to liberty."
9) "When the United States was formed in 1776, it took 19 people on the farm to produce enough food for 20 people. So most of the people had to spend their time and efforts on growing food. Today, it's down to 1% or 2% to produce that food. Now just consider the vast amount of supposed unemployment that was produced by that. But there wasn't really any unemployment produced. What happened was that people who had formerly been tied up working in agriculture were freed by technological developments and improvements to do something else. That enabled us to have a better standard of living and a more extensive range of products."
8) "I want people to take thought about their condition and to recognize that the maintenance of a free society is a very difficult and complicated thing and it requires a self-denying ordinance of the most extreme kind. It requires a willingness to put up with temporary evils on the basis of the subtle and sophisticated understanding that if you step in to do something about them you not only may make them worse, you will spread your tentacles and get bad results elsewhere."
7)“We economists don't know much, but we do know how to create a shortage. If you want to create a shortage of tomatoes, for example, just pass a law that retailers can't sell tomatoes for more than two cents per pound. Instantly you'll have a tomato shortage. It's the same with oil or gas.”
6) “The great virtue of a free market system is that it does not care what color people are; it does not care what their religion is; it only cares whether they can produce something you want to buy. It is the most effective system we have discovered to enable people who hate one another to deal with one another and help one another.”
5) "Workers paying taxes today can derive no assurance from trust funds that they will receive benefits from when they retire. Any assurance derives solely from the willingness of future taxpayers to impose taxes on themselves to pay for benefits that present taxpayers are promising themselves. This one sided 'compact between the generations,’ foisted on generations that cannot give their consent, is a very different thing from a 'trust fund.' It is more like a chain letter."
4) "There are four ways in which you can spend money. You can spend your own money on yourself. When you do that, why then you really watch out what you’re doing, and you try to get the most for your money. Then you can spend your own money on somebody else. For example, I buy a birthday present for someone. Well, then I’m not so careful about the content of the present, but I’m very careful about the cost. Then, I can spend somebody else’s money on myself. And if I spend somebody else’s money on myself, then I’m sure going to have a good lunch! Finally, I can spend somebody else’s money on somebody else. And if I spend somebody else’s money on somebody else, I’m not concerned about how much it is, and I’m not concerned about what I get. And that’s government. And that’s close to 40% of our national income."
3) "Indeed, a major source of objection to a free economy is precisely that it... gives people what they want instead of what a particular group thinks they ought to want. Underlying most arguments against the free market is a lack of belief in freedom itself."
2) "If you put the federal government in charge of the Sahara Desert, in 5 years there'd be a shortage of sand."
1) “I do not believe that the solution to our problem is simply to elect the right people. The important thing is to establish a political climate of opinion which will make it politically profitable for the wrong people to do the right thing. Unless it is politically profitable for the wrong people to do the right thing, the right people will not do the right thing either, or if they try, they will shortly be out of office.”