For Tony for Brendan.
Jim O’Shaughnessy (a cool smart old white guy) who runs Infinite Loops (and a whole bunch of other things) is fond of saying that “We are deterministic thinkers in a probabilistic world.” ……. meaning that…..we don’t have a nice CERTAIN SIMPLE COMPLETE MECHANICAL explanation for things (1) and therefore rely on probability to predict (notice I did not say explain) things (2). This probability thing relies on an assumption called ERGODICITY which, for the vast majority of systems is NOT true. What follows here is a blurb about why this is important….sorta a follow up on why medical literature is so suspect statistically (probabilistically) speaking.
Take the example of smoke filling a room. It fills the room, right? The smoke does not just stay in one corner of the room. Right?
Take the example of mixing a gin and tonic. It tastes sweet and doesn’t taste like straight up navy gin. Right?
Take the example of blow torching a block of metal. The whole block becomes red hot and there is not one spot on the metal that is white hot with the rest is just warm and able to be touched with your finger. Right?
These systems are ERGODIC. It is taken for granted. It is the basis of all (probability) statements about these systems….otherwise known as thermodynamics.
Well…..
What if the smoke stayed in a corner of the room? This would be non-ergodic. Most systems (where humans are involved) ARE NOT ERGODIC! Probability doesn’t work here. But we keep trying. Ergodicity can be thought of in a number of ways.
The Nicholas Nassim Taleb Way
The author of Black Swan fame has called it, “The Most Important Property to Understand in Probability, in Life, in Anything”. His classic example is the turkey. Everything is going along fine. The turkey is fed. The turkey grows. Great. But. Come Thanksgiving, the turkey has his head cut off and is eaten. Game over. There is no nice statistical way to say “one dead turkey”.
Taleb’s other example is with Russian roulette. There is a big difference between six people shooting themselves once and one person shooting himself six times in a row with the roulette gun. If you are the one person (time average) playing the game and pull the trigger six times the game is over on one of those shots because….like…you are freekin’ DEAD.
Economic theory does not really deal with this. Unbelievably, economics does not have any theory of booms and busts.
The Ole Peters Way
This guy is trying to do something about this paradox with something called….yes, you guessed it…..ergodicity economics.
If you are into economics-speak, you can think of ergodicity as the fundamental error in utility theory.
Say you and 99 friends walk into a casino with a collective $100 dollars ($1 each) with a guaranteed strategy that makes 50% every play of the game 99% of the times you play and….a 1% chance of loosing it all. All 100 of you play once and guess what? At the end one round of this completely fabricated non-existent game the collective take is $149 (150 minus 1). This is ensemble probability.
But if you walk into the casino with a hundred bucks and play this same game a hundred times the outcome is different. You might do extraordinarily well for, let’s say, the first 11 goes at the game, but on the 12th go of the game you go broke. The game is over!
Math guys will stay things like, “all of state space is explored” in an ergodic system, followed by some weird looking equation which intimidates most normal people. But math is just another language with all the problems of paradox and rhetoric and bullsh*t that any other language has, whether it’s english, french, or swahili.
Look at this pic.
Pick one of the three squiggly horizontal lines. It (could be a stock market price) goes up and down over time. Now look at those three (it’s more than three actually…it’s N) squiggly lines at the same time (could be three separate stock prices at the same time).
The horizontal line measures the time average. The vertical line measures something call the ensemble average. It they are equal then you have ergodicity.
The Stuart Kauffman Way
The average size (as measure by the number of amino acids stuck together in a long train) of a typical protein in your body is about 200. This train (some are much longer than 200) of amino acids then miraculously folds up to be a real functioning protein in your body. Millions of functioning proteins makes you, the functioning human body. There are ONLY twenty amino acids. The possible number of average proteins of size 200 amino acids is therefore 20 times 20 times twenty and so on…. two hundred times.. right?...each spot has twenty possible choices so the combinatorial of this is twenty to the the 200th Twenty to 200 is ten to the 256th just to let us talk in our more familiar base ten. That is a big number….a̶ v̶e̶r̶y̶ b̶i̶g̶ g̶o̶d̶ d̶a̶m̶n̶e̶d̶ n̶u̶m̶b̶e̶r̶.
Supposedly there are 10 to the 80th particles in the universe. Supposedly the smallest period of time possible is 10 to the minus 44 seconds (called Planck time). And supposedly the universe has been around for about 10 to the 17th seconds (14 billion years, give or take).
So if every particle in the universe made a protein that was 200 amino acids in length every 10 to the minus 44 seconds you would only have 10 to the 141st total number of possible (10 to the 256th) amino acid trains made in this time.
1080+44+17=10141
This is nowhere near the total possible number of proteins which is 10 to the 256th power. So, that means you need a lot of total times that the universe has existed to make all possible proteins……10 to the 115th power in fact.
10256-141=10115
So this amino acid protein thingy has another name. It is the biosphere….you, me, and every other living thing. By this reasoning biology is not ergodic. But this doesn’t stop us from making probabilistic statements about biology. Think about this the next time one of the financial gurus on Bloomberg says the market has such and such a probability of going to such and such a level by such and such date.
Actually Stuart goes on to say that the biosphere above the level of chemistry is non ergodic but I will leave it up to you to read his stuff. It’s really, really good.
The Luca Dellanna Way
Luca’s example is the ski racer. Ergodicity is about survival in a downhill ski race so performance is secondary to survival in the ski race.
Or think about a drunk driver crashing a car. Despite being arrested and losing his license he/she/c̶o̶r̶r̶e̶c̶t̶ p̶r̶o̶n̶o̶u̶n̶s̶ o̶n̶l̶y̶ keeps driving. Dying does stop his driving. BUT….. it is not incentives that provide skin in the game (like Taleb likes to say), but irreversibility.
The stock guys would say that a maximum return is good…unless you go bust….so the smart stock guy goes for that sweet spot between performance and survival.
Have you noticed that this fancy word, ergodicity, is tied up in the theme of our humanity that I can’t stop ranting about….the collective’s rights and responsibilities pitted against those of the individual? There is a fundamental paradox here.
Anyway I hope this rant helps you the next time you hear some medical expert on the TV say, “The vaccine is effective 70% on average”. You will now know there is a big difference between “70% effective in 100% of people” and, “100% effective in 70% of the people”.
Notes
1)Like A→B and B→C…so …A really causes C? Right? Well, yeah in nice simple physical system where we have the necessary information. But the world (us, humanity, biology) ain’t that way. There, more often than not, is an X out there causing C or influencing B…so, as Judea Pearl is fond of saying, “We flounder on the shores of confounding variables.”
2)It is not bad to think of predicting as a future thingy and explaining as a past thingy. If you really want to get into this read "The Best Book I Ever Read".