## Keynes, probability theory and uncertainty

‘The Long Run is a misleading guide to current affairs. In the Long Run we are all dead.’ *– The General Theory of Employment, Interest and Money*

It has become almost a cliché to talk of Keynes’ distinction between risk that can be calculated accurately and what he called ‘irreducible uncertainty’. John Gray in the London Review of Books devotes a long essay to this distinction. And it is equally commonplace for commentators to quote passages in the General Theory, out of context, to suggest that Keynes was inimical to mathematics in general.

But few people now pay attention to one of his most extraordinary books, the ‘Treatise on Probability’, published in 1921. The Treatise (TP) is important for at least two reasons: on the one hand it is, in its own right, and by any measure, a powerful and profound contribution to mathematics and logic, and on the other hand it also plays a central role in his more famous economic theory, especially in relation to his ideas about uncertainty.

What Keynes shows in the TP is that probability can only be calculated to the precision of a single number answer in certain cases. In many more cases it can only be described in terms of what he calls an ‘interval estimate’ between points. So, some probabilities may be calculated as a single number (or ‘point estimate’), a decimal between 0 and 1 where 0 is no probability at all and 1 is what we know to be true (Boolean algebra), but some can only be calculated between ‘bounds’ or ‘intervals’ such as 0.2-0.5 or 0.6-0.8. Keynes further demonstrated that, where two probabilities can only be calculated as intervals and where those intervals overlap, they cannot be compared. Sufficient uncertainty exists about the exact nature of the likelihood of the propositions being true that we cannot tell whether one is more likely than the other.

But Keynes also makes an even more fundamental leap. Instead of defining probability simply as the likelihood of a proposition being true, he introduces a new set of variables. He represents probability as the likelihood of the proposition being true combined with information bearing on the proposition. A relationship he categorizes algebraically as a/h where a is the proposition itself and h is the available information relevant to the proposition. In other words, he describes for the first time in mathematical terms the complex relationship between likelihood of an outcome and the unknown (and perhaps unknowable) information bearing upon this outcome. He criticizes the Bayesian or classical idea of the Principle of Indifference, showing that it holds only when we do not know anything that might impact upon the relative probabilities of an outcome which is apparently equiprobable.

Keynes also saw the importance of ‘non-linearity’ in the calculation of probabilities, and the relationship between this insight and the insights made in the General Theory about economic prediction is all too obvious. Probability need not be continuous, depending on the conditions of our knowledge about a proposition or outcome. Factors that we know nothing about might affect the truth of a proposition or the likelihood of a certain outcome, and unless we know and can calculate the likelihood of those factors with any certainty, we cannot rely on the continuity of probability for the proposition itself.

This may seem obvious, but that is probably my fault for describing it wrongly. In essence, Keynes redefined our understanding of mathematical and statistical probability fifty years before anyone else caught up with him. He used terms no-one else had ever used because he had to invent them to describe his theory. He built on the work of George Boole and presented a strong challenge to the then dominant view of mathematical probability, just as he would do, 15 years later, to the economic consensus.

Not only does this demonstrate that he was a highly sophisticated mathematician, but it also underscore his views about uncertainty, knowledge and risk. Keynes understood that some risks could not be calculated or compared because they relied on assumptions about the future which had no basis in probability, they ignored the non-linear nature of some risks, and they assumed that financial risks, in particular, could be calculated with an actuarial and statistical precision that was simply wrong. Thus the famous passage in the General Theory:

“Too large a proportion of recent “mathematical” economics are mere concoctions, as imprecise as the initial assumptions they rest on, which allow the author to lose sight of the complexities and interdependencies of the real world in a maze of pretentious and unhelpful symbols.” (GT, Book 5, Chapter 21)

Here, Keynes is not criticizing mathematics, or even the application of mathematics to economics *per se. *He is making a more specific point about the liability of recent economic tracts to put mathematics before logic, and to lose the wood for the trees. And he is also, I think, talking very particularly about the models of probabilistic causation used by these writers to predict economic events or uphold economic propositions which, when examined purely in deductive terms, or from experience, prove to be untrue.

In 2008, a large part of the destruction of capital on Wall Street could ultimately be attributed to risk models adopted by the major banks that simply ignored the insights Keynes provided in the *General Theory*. No matter how complicated the equation, some risks will always be incalculable, and some uncertainties will always be irreducible because the future is simply unknowable. But also because factors may bear on these risks that we cannot foresee and have no way of calculating. And because probability may be non-linear or discontinuous in its distribution. In fact, all of this is a vast simplification of Keynes, but if only our bankers and economists today had a millionth of his sophistication, indeed if they had only bothered to read what he wrote, just on this subject quite apart from his more famous works, the global economy would not be in such a dire position today.

I am indebted to Michael Brady’s excellent essay ‘Keynes, Mathematics and Probability: A Reappraisal’