Hedgehogs and foxes

There is a fragment of wisdom that is often attributed to the Ancient Greek poet Archilochus:

“The fox knows many things, but the hedgehog knows one big thing.”

Writing in his essay on Leo Tolstoy, The Hedgehog and the Fox, Isaiah Berlin (pictured) expands Archilochus’ passage into a broad concept of two types of thinkers:

“There exists a great chasm between those, on one side, who relate everything to a single central vision, one system, less or more coherent or articulate, in terms of which they understand, think and feel – a single, universal, organising principle in terms of which alone all that they are and say has significance – and, on the other side, those who pursue many ends, often unrelated and even contradictory, connected, if at all, only in some de facto way, for some psychological or physiological cause, related to no moral or aesthetic principle.”

In The Signal and the Noise, the statistician Nate Silver expanded on this again, and gave some illuminating examples:

“Unless you are a fan of Tolstoy—or of flowery prose—you’ll have no particular reason to read Berlin’s essay. But the basic idea is that writers and thinkers can be divided into two broad categories:

  • Hedgehogs are type A personalities who believe in Big Ideas—in governing principles about the world that behave as though they were physical laws and undergird virtually every interaction in society. Think Karl Marx and class struggle, or Sigmund Freud and the unconscious. Or Malcolm Gladwell and the “tipping point.”
  • Foxes, on the other hand, are scrappy creatures who believe in a plethora of little ideas and in taking a multitude of approaches toward a problem. They tend to be more tolerant of nuance, uncertainty, complexity, and dissenting opinion. If hedgehogs are hunters, always looking out for the big kill, then foxes are gatherers.

Hedgehogs believe in a single, neat, unified approach. Foxes believe in many, messy, disparate approaches.

Or as Karl Weick might put it, hedgehogs and foxes are specialists and generalists.

Confirmation bias

Nassim Nicholas Taleb, writing in The Black Swan:

“Cognitive scientists have studied our natural tendency to look only for corroboration; they call this vulnerability to the collaboration error the confirmation bias.

The first experiment I know of concerning this phenomenon was done by the psychologist P. C. Wason. He presented the subjects with the three number sequence 2, 4, 6, and asked them to try to guess the rule generating it. There method of getting us to produce other three number sequences, to which the experimenter would respond “yes” or “No” depending on whether the new sequences were consistent with the rule. Once confident with their answers, the subjects would formulate the rule. … with the correct rule was “numbers in ascending order,” nothing more. Very few subjects discovered it because in order to do so I had to offer a series in descending order (what the experimenter word say “no” to). Wason noticed that the subjects had a role in mind, but gave him examples aimed at confirming it instead of trying to supply series that were inconsistent with their hypothesis. Subjects tenaciously kept trying to confirm the rules that they had made up.”

Peter Cathcart Wason‘s 2-4-6 problem proved that we have a tendency to seek out evidence which confirms our ideas rather than that which falsifies them.

Alvin Toffler echoed a sentiment, and borrowed a fair amount from Leon Ferstinger‘s concept of cognitive dissonance, 10 years later.

Nate Silver writing in The Signal and The Noise:

“Alvin Toffler, writing in the book Future Shock in 1970, predicted some of the consequences of what he called “information overload”. He thought our defence mechanism would be to simplify the world in ways that confirmed our biases, even as the world itself was growing more diverse and more complex.”

Silver later sums up the bias succinctly:

“The instinctual shortcut that we take when we have ‘too much information’ is to engage with it selectively, picking out the parts we like and ignoring the remainder.”

Rational bias

Nate Silver writing in The Signal and The Noise:

“When you have your name attached to a prediction your incentives may change. For instance, if you work for a poorly known firm, it may be quite rational for you to make some wild forecasts that will draw big attention when they happen to be right, even if they aren’t going to be right very often. Firms like Goldman Sachs, on the other hand, might be more conservative in order to stay within the consensus.

Indeed, this exact property has been identified in the blue chip forecasts: one study terms the phenomenon “rational bias”. The less reputation you have, the less you have to lose by taking a big risk when you make a prediction. Even if you know the forecast is dodgy, it might be rational for you to go after the big score. Conversely, if you have already established a good reputation, you might be reluctant to step too far far out of line even when you think the data demands it.”

The greater your reputation, the more conservative you are.

Theory of complexity

In his book Black Swan, the essayist Nicholas Nassim Taleb provides a functional definition of complex domains:

“A complex domain is characterised by the following: there is a great degree of independence between its elements, both temporal (a variable depends on its past changes), horizontal (variables depend on one another), and diagonal (variable A depends on the past history of variable B). As a result of this independence, mechanisms are subjected to positive, reinforcing feedback loops.”

Nate Silver expands on this brief introduction with a more illustrative description in his book The Signal and the Noise:

“The theory of complexity that the late physicist Per Bak and others developed is different from chaos theory, although the two are often lumped together. Instead, the theory suggests that a very simple things can behave in strange and mysterious ways when they interact with one another.

Bak’s favourite example was that of a sandpile on the beach. If you drop another grain of sand onto the pile (…) it can actually do one of three things. Depending on the shape and size of the pile, it might stay more or less where it lands, or it might cascade gently down the small hill towards the bottom of the pile. Or it might do something else: if the pile is too steep, it could destabilise the entire system and trigger a sand avalanche.”

Just imagine the number of different ways that the sandpile could be configured. And just imagine the number of ways the falling grain of sand could hit the pile. Despite being such a simple object (a sandpile) the number of possible interactions between its constituent parts are innumerable. And each potential scenario would have a different result.

But of course a pile of sand containing thousands of irregular grains is complex. An simpler example would be the initial break in a game of pool. 16 spheres on a flat surface. But still, how many times would you have to break until every ball landed in the exact same positions?

These are complex systems.

Whilst Silver is quick to distinguish between complexity and chaos, it’s worth noting that Tim Harford is also keen to make a distinction. In his book Adapt he separates the concepts of complex systems and tightly coupled systems.

To put it simple, complex systems have a lot of possible, hard to predict scenarios. Some will destabilise the entire system, some won’t. Tightly coupled systems are always the latter.

Known unknowns

During a press conference in 2002 a reporter questioned Donald Rumsfeld, the then US Secretary of Defence, about the presence of weapons of mass destruction in Iraq.

Rumsfeld’s response included the famous line:

“There are known knowns; there are things we know we know. We also know there are known unknowns; that is to say we know there are some things we do not know. But there are also unknown unknowns; there are things we do not know we don’t know.”

Unknown unknowns are a major problem when one is seeking to understand and asses a situation. You don’t know the critical piece of information and you don’t know that you need to know it.

Nate Silver discusses this idea in The Signal and the Noise and seeks to clarify the terms use:

“The concept of the unknown unknown is sometimes misunderstood. It’s common to see the term employed in formulations like this, to refer to a fairly specific (but hard-to predict) threat:

Nigeria is a good bet for a crisis in the not-too-distant future—an unknown unknown that poses the most profound implications for US and global security [emphasis added].

This particular prophecy about the terrorist threat posed by Nigeria was rather prescient (it was written in 2006, three years before the Nigerian national Umar Farouk Abdulmutallab tried to detonate explosives hidden in his underwear while aboard a flight from Amsterdam to Detroit). However, it got the semantics wrong. Anytime you are able to enumerate a dangerous or unpredictable element, you are expressing a known unknown. To articulate what you don’t know is a mark of progress.

Few things, as we have found, fall squarely into the binary categories of the predictable and the unpredictable. Even if you don’t know to predict something with 100 percent certainty, you may be able to come up with an estimate or a forecast of the threat. It may be a sharp estimate or a crude one, an accurate forecast or an inaccurate one, a smart one or a dumb one.* But at least you are alert to the problem and you can usually get somewhere: we don’t know exactly how much of a terrorist threat Nigeria may pose to us, for instance, but it is probably a bigger threat than Luxembourg.”

Wealth effect

Nate Silver describes the wealth effect in The Signal and the Noise:

“Nonhousehold wealth – meaning the sum total of things like savings, stock, pensions, cash, and equity in small businesses – declined by 14% for the median family between 2001 and 2007. When the collapse of the housing bubble whiped essentially all their housing equity of the books, middle-class Americans found they were considerably worse off than they had been a few years earlier.

The decline in consumer spending that resulted as consumers came to take a more realistic view of their finances – what economists call a “wealth effect” – is variously estimated at between 1.5% and 3.5% of GDP per year, potentially enough to turn average growth into a recession.”

When people believe themselves to be wealthier, they tend to spend more money.

This rule of thumb seems to hold true even when wealth is tied up in assets.

For example, although a rise in the value of your property doesn’t provide you with more cash right now, it does seem to cause you to spend more.

Alternative, although a fall in the value of your stock portfolio doesn’t reduce the amount of money available to you right now, it does seem to cause you to spend less


Chaos theory

The following extract is taken from Nate Silver‘s book on the art and science of prediction, ‘The Signal and the Noise’.

“You may have heard the expression: the flap of a butterfly’s wings in Brazil can set off a tornado in Texas . It comes from the title of a paper delivered in 1972 by MIT’s Edward Lorenz, who began his career as a meteorologist.”


“Lorenz and his team were working to develop a weather forecasting program on an early computer known as a Royal McBee LGP-30. They thought they were getting somewhere until the computer started spitting out erratic results. They began with what they thought was exactly the same data and ran what they thought was exactly the same code—but the program would forecast clear skies over Kansas in one run, and a thunderstorm in the next.

After spending weeks double-checking their hardware and trying to debug their program, Lorenz and his team eventually discovered that their data wasn’t exactly the same: one of their technicians had truncated it in the third decimal place. Instead of having the barometric pressure in one corner of their grid read 29.5168, for example, it might instead read 29.517. Surely this couldn’t make that much difference?

Lorenz realized that it could. The most basic tenet of chaos theory is that a small change in initial conditions—a butterfly flapping its wings in Brazil—can produce a large and unexpected divergence in outcomes—a tornado in Texas. This does not mean that the behaviour of the system is random, as the term “chaos” might seem to imply. Nor is chaos theory some modern recitation of Murphy’s Law (“whatever can go wrong will go wrong”). It just means that certain types of systems are very hard to predict.

The problem begins when there are inaccuracies in our data. (…). Imagine that we’re supposed to be taking the sum of 5 and 5, but we keyed in the second number wrong. Instead of adding 5 and 5, we add 5 and 6. That will give us an answer of 11 when what we really want is 10. We’ll be wrong, but not by much: addition, as a linear operation, is pretty forgiving. Exponential operations, however, extract a lot more punishment when there are inaccuracies in our data. If instead of taking 55—which should be 3,215—we instead take 56 (five to the sixth power), we wind up with an answer of 15,625. That’s way off: we’ve missed our target by 500 percent.

This inaccuracy quickly gets worse if the process is dynamic, meaning that our outputs at one stage of the process become our inputs in the next. For instance, say that we’re supposed to take five to the fifth power, and then take whatever result we get and apply it to the fifth power again. If we’d made the error described above, and substituted a 6 for the second 5, our results will now be off by a factor of more than 3,000.22 Our small, seemingly trivial mistake keeps getting larger and larger.”

Silver summarises,

“Chaos theory applies to systems in which each of two properties hold:

  • The systems are dynamic, meaning that the behavior of the system at one point in time influences its behavior in the future;
  • And they are nonlinear, meaning they abide by exponential rather than additive relationships.”

When systems are dynamic and non-liner, small changes compound themselves, and keep compounding themselves exponentially.

In tightly coupled systems, an event has knock-on effects; like the toppling of the first domino. In chaotic systems for every domino that falls, 2 get added to the line.

Okun’s Law

The following extract is taken from The Signal and the Noise by the American statistician, Nate Silver. In it he describes the economic relationship between job growth and GDP growth as first proposed by Arthur Melvin Okun in 1962.

“The American and global economy is always evolving, and the relationships between different economic variables can change over the course of time.

Historically, for instance, there has been a reasonably strong correlation between GDP growth and job growth. Economists refer to this as Okun’s Law. During the Long Boom of 1947 through 1999, The rate of job growth had normally been about half the rate of GDP growth, so if GDP increased by 4% during a year, the number of jobs would increase by about 2%.

The relationship still exists – more gross is certainly better for jobseekers. But it’s dynamics seems to have changed. After each of the last couple of recessions, considerably fewer jobs were created than would have been expected during the Long Boom years. In the year after the stimulus package was passed in 2009, for instance, GDP was growing fast enough to create about two million jobs according to Okun’s law. Instead, an additional 3.5 million jobs were lost during the period.”

Bayes’s Theorem

The following extract is taken from Nate Silver’s book “The Signal and the Noise”.

Thomas Bayes’s paper, “‘An Essay toward Solving a Problem in the Doctrine of Chances’, was not published until after his death, when it was brought to the Royal Society’s attention in 1763 by a friend of his named Richard Price. It concerned how we formulate probabilistic beliefs about the world when we encounter new data.

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