Uncertainty in economics and in life

"Doubt is not a pleasant condition, but certainty is an absurd one." -- Voltaire, quoted in Mervyn King, The End of Alchemy.

"At the heart of modern macroeconomics is the same illusion that uncertainty can be confined to the mathematical manipulation of known probabilities." -- Mervyn King, ibid. p.121.

As has been argued elsewhere, for example in Taleb's Black Swan and in Mandelbrot et. al. The Misbehaviour of Markets, many market analysts, economists, politicians and others, including probably most of us ordinary folk, imagine that most things stay more-or-less the same most of the time, and when things change they do so within reasonable boundaries. Those of a mathematical persuasion may further imagine that things follow a normal or Gaussian distribution, that is, events are distributed in a bell-curve, in which very few things are extreme and most things are more-or-less average.

This is the case with things like people's height, for example, or fitness, or biological variables (see my earlier blog post about the misuse of this) or workers' incomes (excluding oil billionaires and bank executives). However it is not the case with non-linear dynamic systems.

What, I hear you ask, is a non-linear dynamic system? It is any system in which there is any degree of complexity in the relation between inputs and outputs, and in which the outputs feed back as inputs. An example is the weather and the so-called butterfly effect, in which the flapping of a butterfly's wings can change the course of a tornado several weeks hence. An outcome can be radically changed by a small change in initial conditions. This is why even with much more powerful computers it is unlikely that the weather will be predictable with any accuracy more than ten days ahead, because it is simply not feasible that we could ever measure the initial conditions with sufficient accuracy.

When we come to the stock market there are so many variables and so many unknowns that long-term prediction is impossible. Someone bets on an oil company, then a major accident happens and the stock falls, or else someone announces a major breakthrough in solar power and the stock falls, or there is a war involving a major oil exporter and the price of oil rises. I such cases the unexpected occurs frequently and is better represented by some kind of fat-tailed distribution, in which uncommon events happen more frequently.

There are up-sides and down-sides to this. Our lives in general are subject to the unexpected. We enjoy the stability of relationships and our homes and jobs and we value living in a country that is not at war, if we think about it at all. But the unexpected does happen. We prefer not to think about it, other than the approximately 1 in 14 million chance that our lottery ticket will win.

The up-side of the butterfly effect, though, is that if the beating of the wings of a butterfly in China can change the course of a tornado on the other side of the world, imagine what the effect of a smile could be!

An interesting medical scam

When I was writing the chapter about the pink alternative medicine clinic, I found myself challenged to come up with something so bizarre that it would be obvious it was a satire. Just so, I later discovered that bananas occasionally emit positrons. This means that the wobbly bananas I described are occasionally sending out electrons that are going backwards in time.

No doubt sooner or later someone in the alternative medicine world will conclude that sufficient exposure to wobbly bananas, preferably while lying naked on a couch, can make you younger. Of course you'd have to be properly quantumly entangled with the bananas.

However there are some forms of medical wackyology that are more cleverly dressed in the outer garments of science and can fool even quite intelligent people. I here present one such piece of tomfoolery for your education.

It works like this: the practitioner promises to improve the patient's health by re-balancing them. To do this he or she orders a very large number of blood tests. Then, if there are any blood tests that are abnormal the practitioner will offer to rectify this by prescribing some supplement or other.

Sounds reasonable, doesn't it? After all, if someone feels tired and their doctor finds out that they have a deficiency of (say) iron, they can fix it by prescribing iron tablets.

However, medicine is generally a bit more complicated than that. If someone is genuinely iron-deficient it is important to ask why, and to look for the cause. In this context, merely 're-balancing' could be dangerously negligent.

The real basis of the scam is actually mathematical. Consider a field full of women of the same age. Line them up according to height. Most of them will be somewhere around the average, but there will be a few really short ones and a few really tall ones. Even if we remove the ones who are tall or short because of some medical abnormality, there will still be a variation. Thus, healthy people can be tall or short, and a few will be extra tall or extra short without there being anything wrong with them.

(Image taken from this web site)

When the laboratory measures something in someone's blood, most of the time the result will be somewhere in the middle of the range, but some results will be outliers. This will happen fairly often even in perfectly healthy people. There is no absolute cut-off where something suddenly becomes abnormal. The laboratory somewhat arbitrarily defines the range as the middle 95% of values. This should not be referred to as 'the normal range,' rather, it is properly referred to as 'the laboratory reference range.'

It's the same thing with height. If a child is in the bottom 10% for height, you don't think much about it, especially if their parents are short too. If they are in the bottom 2% for height, you start wondering if there is a problem, and you look at the parents' heights and at the previous measurements of growth to see if there is a pattern suggesting disease or not.

In the same way, if a laboratory value is outside the laboratory reference range, you ask yourself if there is a pattern to it or not (such as, all the liver tests are out and the patient drinks too much, or, only one of them is out and the patient is a perfectly healthy teetotaller).

So now we come to the essence of the scam. One or two values being 'out' may or may not indicate a disease process. What is the probability that at least one value will be out in a perfectly healthy person?

Consider first the simpler case of tossing a coin. What is the probability that it will be heads? Fairly obviously 1 in 2, or half, or as normally expressed, a probability of 0.5.

What is the probability that it will be heads twice in two tosses? The answer is the probability that it will be heads the first time multiplied by the probability that it will be heads the second time: 0.5 x 0.5 = 0.25 (a quarter). Three times in three tosses? 0.5 x 0.5 x 0.5 = 0.125 (an eighth) - and so on. What is the probability that in three tosses of a coin at least one will be tails? 1 - 0.125 = 0.875 (seven-eighths).

To do our calculation we use the same process. The probability of a test being within the laboratory reference range is 0.95 (19 times out of 20). The probability of  two tests being within the reference range is 0.95 x 0.95 = 0.9025 (nearer 18 times out of 20). We multiply the probability together with itself the number of times there are tests. If there are 20 tests that is 0.95^20=0.36. This is the probability that all the tests will be 'normal' - just over a third. To get the probability of at least one test being 'out' we subtract this number from 1. So if we do 20 tests on a healthy person, the probability that at least one will be out of range is almost two-thirds (0.64).

For thirty tests the probability of at least one test being 'out' is 0.78, or more than three-quarters. So in such a case on average three out of four healthy people will be convinced they need 're-balancing' and will no doubt purchase whatever is recommended to do the re-balancing, plus of course the consultation fee. Of course this is unlikely to make any difference to anything so they'll be back for more until they decide to save their money and get on with the rest of their lives.

I'm not a cynic, I am simply reporting what I see. As the saying goes, 'you do the math.'