Why are some systems harder to predict?
Science is fundamentally about identifying cause and effect to be able to make predictions. However, different systems differ enormously in how easily this can be done. To explain why, in 1948 Warren Weaver proposed all systems fell into one of three types.
First there are systems like snooker. If you know how fast one ball is going and where it hit another, Newtonian mechanics can predict exactly what will happen, meaning you can map every cause to its effect. We call this simple deterministic causality.
Next imagine hundreds of snooker balls on a giant table. You can’t follow every collision, but it turns out for some systems you don’t need to.
Knowing just how many balls there are and how much energy they have between them, we can use statistics to predict how many will be moving at what speeds. We call this disconnected causality because no single collision has a lasting effect on the whole, so knock any one ball off course and the overall spread of speeds stays the same. We also say these balls interact linearly, because the effects of individual collisions simply add up and average out, so the whole behaves like the sum of its parts.
Then finally we have what we have come to call complex systems.
By 1926 hunters had killed off the last wolves in Yellowstone National Park. With no predators to hassle them, the park’s elk and deer spent more time by the rivers, eating the young willows and aspens along the banks. Without those trees holding the soil together the banks eroded, and the rivers began to change course. It’s unclear how far the chain reached, but what is clear is by simply removing one predator the whole landscape began to reshape itself.
Unlike the giant snooker table, you couldn’t have predicted this from a few high-level measures like the number of animals or the average rainfall. In these systems the details matter.
This is because a small change in one part cascades and amplifies into large changes somewhere else. We say these systems have intertwined causality or non-linear interactions.
Most of the statistics we’ve developed so far assumes a system sits at the snooker end, where a few averages describe the whole. This is a practical assumption, given it’s the maths we know well and have available to us, but it’s also a dangerous assumption.
The tricky part is no system is simply complex or not, though. Which means we need a measure of how complex a system is, so we can know when those averages can be trusted.
How do we define complexity?
The cleanest way into this is a thought experiment from the physicist Sean Carroll. Pour cream onto black coffee and at first the two sit in separate layers. You can describe this in five words: cream on top, coffee below. Start stirring and the cream forms into a really interesting set of complex swirls and streaks. Where describing these shapes in any sort of detail would take ages. Until finally if you keep stirring, everything blends into one uniformly mixed liquid, which is again extremely easy to describe.
Carroll leads us to a working definition of complexity being roughly how hard something is to describe. Which put another way is asking how much the details matter. The layered cup and the mixed cup are both simple, because a single sentence tells you everything about them. Only the swirly cup in the middle needs its details, in the same way we needed to understand the roles of the wolves in Yellowstone.
As you stir a coffee we say its entropy increases. Carroll describes the coffee starting out as low entropy because it’s nicely organised. Then slowly increases as the two liquids become fully mixed. The odd thing is for this system, complexity seems to only appear for the middle swirly section.
There have been many attempts to define complexity, but none that were that successful at actually measuring it. So in 2012 the complexity scientists Carlos Gershenson and Nelson Fernández turned the coffee cup into a measure. They made a simple construct, where you score a system as zero complexity when its entropy is at its lowest. Then zero again when it’s at its highest. Defining a measure that peaks exactly halfway between, just like for the swirly cup.
It was an important step forward, but as the rest of this article shows, it wasn’t enough on its own.
What is entropy and information?
To see why, we need to be completely clear what entropy is and how it links to information.
Imagine a closed box. It has four spaces and holds four coloured balls. What you know from the outside is called the macro state (four balls, four spaces). Each way the balls could be arranged inside is called the micro state. For this box there are 24 of them.
Now picture a bigger box with 16 spaces with the same four balls inside. Because the balls have so many more places they could be, the number of possible arrangements, or micro states, jumps from 24 to 43,680.
Entropy measures how many micro states fit the macro state, which is the same as asking how uncertain you are about how it’s arranged inside. Guessing the setup of the small box is relatively easy (1 in 24). But in the big box it’s much harder (1 in 43,680). We say the bigger box has the higher entropy, not because anything inside it is messier, but because knowing the big picture tells you less about the details.
The same idea works for anything you can make guesses about, including text. If you’ve just read “AAAAAAAAAA” you’d have a high confidence about what the next letter is likely to be. However, guessing what comes after “ABJSTOMXGW” is much harder, so we say that string has a higher entropy. This is also why entropy measures information. With the first string seeing yet another A tells you practically nothing you hadn’t already guessed. Meanwhile whatever the next letter of the random string is is likely a surprise. And being surprised means you didn’t predict it, which means you got some new information (compared to the A you had guessed).
The problem
We previously said that complexity was simply in the middle between low entropy and high entropy. However, let’s go back to the snooker table. If we suddenly made it bigger its entropy would definitely rise but does the system change? Would that snooker table momentarily become complex?
No. The balls still interact in exactly the same way they did before.
As we’ve seen, complexity is about causality, where a small change can lead to a big one. So the question becomes how do we tease apart what is happening in the coffee cups from snooker balls?
The solution
The trick is that entropy is all about the micro states, and we need to look at the collection of them (known as an ensemble). No matter how the balls are arranged, each time you expand the table, the way they spread will look (at a macro level) basically identical.
Now do the same with the coffee. Before stirring, every cup has the same neat layer of cream on top, so if you overlay them all, the average cup looks just like each other one. Once they’re fully mixed the same is true, since every cup is the same light brown liquid and so is their average.
The middle is different. When we overlay them, the swirls blur into each other. The average looks much more like the fully mixed cup than like any of the cups that made it.
A more mixed cup has a higher entropy. This is the key insight: complex averages have higher entropies. So we can subtract the entropy of the individual cups from the entropy of the averaged cup and measure exactly how much a system is complex. The paper covers the exact details, but this is the broad idea.
A key characteristic of complexity is whether the average of a system can accurately represent a typical instance of that system.
Practical example
Say you’re crossing the country by train to get to a wedding. There are two trains, each taking an hour, with a 10-minute change in the middle, and you need a taxi waiting at the other end, so you have to tell the driver when you’ll arrive.
On paper the journey takes 2 hours 10 minutes. The first train is usually a few minutes late, though, and if it’s more than 10 minutes late you miss the connection and wait an hour for the next one. On this line that happens about two times in five.
That makes this a complex system, even though nothing about it is complicated. A first train that is 9 minutes late and one that is 11 minutes late are almost the same train, but one gets you there at 2 hours 10 and the other at 3 hours 10, so two minutes at the start become an hour at the end.
Run the journey enough times and the arrival times pile up into two separate peaks, a tall one at 2 hours 10 minutes and a shorter one at 3 hours 10. The average comes to 2 hours 34 minutes, which lands in the empty gap between them.
The average journey time is one that almost never actually happens.
You know instinctively it would be stupid to book a taxi to pick you up based on the average journey time. You’d either be needlessly waiting around on the platform for 24 minutes, or the driver would be waiting around for 36 minutes for you. The optimal strategy is to be hopeful and book for 2 hours 10, the time you’ll arrive if all goes well, then rebook if you miss the connection.
This is a simple example, and so one most of us will get right through experience. However, in more complex situations it’s surprising how often we rely on statistics which don’t apply.
Real example
In the late 1940s the US Air Force had a serious problem: its pilots kept losing control of their planes. This was the start of the jet age, so the aircraft were faster and harder to fly than anything before, and the accidents piled up. At the worst point, 17 pilots crashed in a single day.
The first suspect was the pilots themselves, and many crashes were put down to pilot error. But the pilots were experienced, the training checked out, and when engineers went over the controls and electronics they found nothing wrong. So attention turned to the one thing nobody had questioned, the cockpit. Its layout dated from 1926, when engineers measured hundreds of pilots and built the seat, pedals and controls around their average. Perhaps, the thinking went, pilots had simply grown since then.
In 1950 researchers at Wright Air Force Base in Ohio set out to find the new average. They took 140 measurements of 4,063 pilots, from thumb length and crotch height to the distance between eye and ear. On the team was Lt. Gilbert Daniels, who had studied physical anthropology at Harvard. For his undergraduate thesis he had measured the hands of 250 students from very similar backgrounds and found that their hands were not similar at all, so he had his doubts about the average pilot.
Daniels took the ten measurements that matter most for a cockpit, such as height, chest size and sleeve length, and asked how many pilots were average on all ten. He was generous about what counted as average, allowing anyone in the middle 30% of the range for each measurement, and most of his colleagues expected the bulk of pilots to qualify. The answer was zero. Not one of the 4,063 pilots was average on all ten, and even on just three measurements fewer than 3.5% were.
The cockpit built for the average pilot had fitted no pilot at all. “The tendency to think in terms of the ‘average man’ is a pitfall into which many persons blunder,” Daniels wrote in his 1952 report. The human body is complex: your arm length is related to your leg length, but only loosely, so being average in one way tells you very little about the rest of you. Just like the taxi booked for the average journey, a design built for the average ends up suiting almost nobody.
The Air Force’s answer was to stop designing for the average. It told manufacturers that cockpits had to fit every pilot from the 5th to the 95th percentile on every measurement. The manufacturers said this would be too expensive, but the fixes turned out to be cheap: adjustable seats, adjustable foot pedals, and adjustable helmet straps and flight suits. Pilot performance improved, and the same idea is why the seat in your car slides back and forth today. The best cockpit, it turned out, was an adjustable one.
Business example
Say you run a silicon chip factory and you’re choosing between two expensive upgrades. You can only afford to build one, so before you commit you commission a digital twin of the factory, a simulation detailed enough to tell you how many minutes each chip will take to make under each option. Today it takes about 100.
The tempting thing is to run the twin once for each option. If you had, Option B might well have come back at 83 minutes, a 17% improvement, and you would have signed off on it that afternoon. You would only have found out after building it what the result usually is.
So you do the careful thing and run each option five times, then compare them the way any analyst would, by the average and the spread. Option B averages 94 minutes against Option A’s 96, and its spread is tighter too. Every standard number points to B.
Now look at the individual runs. Option A’s five runs sit almost on top of each other, so whichever one turns out to be the real factory, you get about 96 minutes. Option B’s runs disagree with each other. One is brilliant, and the other four land between 93 and 99.5, which is no better than A and at worst barely better than what you have today. Its average of 94 falls in a gap that none of the runs actually produced, just like the average journey that no train passenger ever takes.
This matters because you only get to build one factory, which means you only get one run. Choosing B is a one-in-five bet on a big win, and otherwise you’ve paid for an upgrade that delivers what A would have, or less. When customers are counting on the chips you’ve promised them, knowing what you’ll get is worth far more than a couple of minutes on paper, so Option A is the better choice.
Traditional statistical tests, such as ANOVA or chi-squared, would flag that Option B’s runs differ, but that is all they do. They are built to decide whether runs differ, not how much, so they hand you a confidence level rather than a size you can weigh against two minutes on paper. The same problem helps explain why the replication crisis has hit complex fields like psychology so hard: a single study is a single run, and in an incoherent system the next run can tell a different story.
Taking the average and variance is necessary but not sufficient.
Read on: self-organisation
Incoherence tells you when the average stops describing a typical run. The natural next question is whether those runs fall into a handful of recognisable patterns, which is what we mean by self-organisation. That is the subject of the follow-up explainer, Self-organisation, and how to measure it, which introduces a second measure called Cohesion and shows how it links to Bayesian statistics in the same way Incoherence links to frequentist statistics.
Naming it Incoherence
We call this Incoherence. First, because complexity is such an overloaded term. But also because to be coherent means being consistent and understandable. This is echoed in physics, where coherent waves align with each other.
Why a new measure?
Statistics already has tests like ANOVA and chi-squared that tell you when a set of runs differ from each other, so it’s fair to ask why we need anything new. Well first, those tests start from the assumption that the runs are all the same, and only once you’ve gathered enough data will they say otherwise. That makes sense when you’re testing whether a new drug works, because the burden of proof should be on the drug. For complex systems it’s the wrong way round. Treating a complex system as simple costs far more than the reverse. Just as importantly, what the metrics give you is how confident you can be in the disagreement, not a measure of how much they disagree.
The closer relative is the Jensen–Shannon divergence, which Incoherence is built on and which uses the same subtraction we did with the coffee cups. In the paper I tested the two side by side, and the divergence is the wrong tool for the job in a few ways. First its value shifts with the number of possible outcomes, so it can’t be compared across systems. Second it measures disagreement in absolute terms, where Incoherence measures it relative to the entropy of the pooled runs, so disagreement counts for more in a system that ought to be orderly. And finally it averages the gaps between each run and the pool, where Incoherence squares them as variance does, because catching black swans and outliers matters.
Most importantly though, a better test would just be one more tool. What Incoherence does is update how we tie information to complexity: the information you lose when the average fails to describe any instance is a measurable output of the complexity of the system.
It does have limitations though. Obviously this is not a definition of what causes complexity (that’s the non-linear interactions), it’s just a measure of it. It also requires breaking a system into an ensemble, whether that’s the same system over many simulations, the global economy split into countries, or a set of snowflakes compared with each other. And it relies on entropy, which although extremely flexible comes with a lot of subjectivity as to what gets measured.
The link with frequentist statistics
Most of the statistics we learn at school is frequentist. It treats probability as a long-run frequency: a fair coin is one that lands heads half the time, and you find that out by flipping it a lot and counting. The train journeys and the factory’s digital twin both worked this way, by pooling the repeats and reading off the average.
Pooling only describes a single repeat if every repeat is a draw from the same thing. Flipping one coin 100 times tells you the same as flipping 100 coins once each, because every coin behaves like every other. The factory’s five runs are not like that. Each is a different factory you might end up with, and once you build it you only get that one, making chip after chip. Pool the five and the average is 94 minutes, which is true of the five taken together but of no factory you could actually build. You can’t get round this by testing, either. With a coin you can flip it a few times before you bet, but by the time you can measure the factory you have already paid for it.
Incoherence measures how far the runs are from being draws from the same thing. Each run has its own spread of chip times, and so its own entropy, and the pool of all five has an entropy too. Incoherence is the gap between them, the same subtraction we did with the coffee cups. Option A’s runs each look like the pool, so the gap is close to zero and the pooled average of 96 is a fair forecast of the factory you would get. Option B’s runs are narrow and sit in different places while the pool stretches across all of them, so the gap is large and its average of 94 describes none of them. Every number a frequentist takes from the pool rests on the runs being alike, and Incoherence measures how far that holds.
Most real systems sit somewhere in between, like the trains. A year of commuting tells you something about the one journey you take on the wedding day, but not everything, and Incoherence puts a number on how much.
Incoherence tells you how much you can rely on the frequentist interpretation.
Thanks
I want to give particular thanks to Carlos Gershenson, who first proposed the 2012 measure of complexity that this work builds on. Carlos also generously edited the special issue of Entropy in which the Incoherence paper was published.