A Treatise on Probability

A Treatise on Probability, published by John Maynard Keynes in 1921, provides a much more general logic of uncertainty than the more familiar and straightforward 'classical' theories of probability. This has since become known as a "logical-relationist" approach, and become regarded as the seminal and still classic account of the logical interpretation of probability (or probabilistic logic), a view of probability that has been continued by such later works as Carnap's Logical Foundations of Probability and E.T. Jaynes Probability Theory: The Logic of Science.

Keynes's conception of this generalised notion of probability is that it is a strictly logical relation between evidence and hypothesis, a degree of partial implication. It was in part pre-empted by Bertrand Russell's use of an unpublished version.

In a 1922 review, Bertrand Russell, the co-author of Principia Mathematica, called it "undoubtedly the most important work on probability that has appeared for a very long time," and said that the "book as a whole is one which it is impossible to praise too highly."

With recent developments in machine learning to enable 'artificial intelligence' and behavioural economics the need for a logical approach that neither assumes some unattainable 'objectivity' nor relies on the subjective views of its designers or policy-makers has become more appreciated, and there has been a renewed interest in Keynes's work.

Here Keynes generalises the conventional concept of numerical probabilities to expressions of uncertainty that are not necessarily quantifiable or even comparable.

In Chapter 1 'The Meaning of Probability' Keynes notes that one needs to consider the probability of propositions, not events.

In Chapter 2 'Probability in Relation to the Theory of knowledge' Keynes considers 'knowledge', 'rational belief' and 'argument' in relation to probability.

In Chapter 3 'The Measurement of Probabilities' he considers probability as a not necessarily precise normalised measure and used the example of taking an umbrella in case of rain to illustrate this idea, that generalised probabilities can't always be compared.

Chapter 4 'The Principle of Indifference' summarises and develops some objections to the over-use of 'the principle of indifference' (otherwise known as 'the principle of insufficient reason') to justify treating some probabilities as necessarily equal.

In Chapter 5 'Other Methods of Determining Probabilities' Keynes gives some examples of common fallacies, including:

He also presents some arguments to justify the use of 'direct judgement' to determine that one probability is greater than another in particular cases.

Chapter 6 'Weight of Argument' develops the idea of 'weight of argument' from chapter 3 and discusses the relevance of the 'amount' of evidence in support of a given probability judgement. Chapter 3 further noted the importance of the 'weight' of evidence in addition to any probability:

Chapter 7 provides a 'Historical Retrospect' while Chapter 8 describes 'The Frequency Theory of Probability', noting some limitations and caveats. In particular, he notes difficulties in establishing 'relevance' and, further, the lack of support that the theory gives for common uses of induction and statistics.

Part 1 concludes with Chapter 9 'The Constructive Theory of Part I. Summarised.' Keynes notes the ground to be covered by the subsequent parts.

This part has been likened to an appendix to Russell and Whitehead's Principia Mathematica. According to Whitehead Chapter 12 'The Definition and Axioms of Inference and Probability'

Chapter 14 'The Fundamental Theorems of Probable Inference' gives the main results on the addition, multiplication independence and relevance of conditional probabilities, leading up to an exposition of the 'Inverse principle' (now known as Bayes Rule) incorporating some previously unpublished work from W. E. Johnson correcting some common text-book errors in formulation and fallacies in interpretation, including 'the fallacy of the middle term'.

In chapter 15 'Numerical Measurement and Approximation of Probabilities' Keynes develops the formalism of interval estimates as examples of generalised probabilities: Intervals that overlap are not greater than, less than or equal to each other.

Part 2 concludes with Chapter 17 'Some Problems in Inverse Probability, including Averages'. Keynes' concept of probability is significantly more subject to variation with evidence than the more conventional quantified classical probability.

Here Keynes considers under what circumstances conventional inductive reasoning might be applicable to both conventional and generalise probabilities, and how the results might be interpreted. He concludes that inductive arguments only affirm that 'relative to certain evidence there is a probability in its favour'.

Chapter 21 'The Nature of Inductive Argument Continued' discusses the practical application of induction, particularly within the sciences.

Part 3 concludes with Chapter 23 'Some Historical Notes on Induction'. This notes that Francis Bacon and John Stuart Mill had implicitly made assumptions similar to those Keynes criticised above, but that nevertheless their arguments provide useful insights.

Here Keynes considers some broader issues of application and interpretation. He concludes this part with Chapter 26 'The Application of Probability to Conduct'. Here Keynes notes that the conventional notion of utility as 'mathematical expectation' (summing value times probability) is derived from gambling. he doubts that value is 'subject to the laws of arithmetic' and in any case cites part 1 as denying that probabilities are. He further notes that often 'weights' are relevant and that in any case it 'assumes that an even chance of heaven or hell is precisely as much to be desired as the certain attainment of a state of mediocrity'. He goes on to expand on these objections to what is known by economists as the expected utility hypothesis, particularly with regard to extreme cases.

Keynes ends by noting:

and

Keynes goes beyond induction to consider statistical inference, particularly as then used by the sciences.

In Chapter 28 'The Law of Great Numbers' Keynes attributes to Poisson the view that 'in the long ... each class of events does eventually occur in a definite proportion of cases.' He goes on:

The key chapter is Chapter 32 'The Inductive Use of Statistical Frequencies for the Determination of Probability a posteriori - The Method of Lexis'. After citing Lexis' observations on both 'subnormal' and 'supernormal' dispersion, he notes that 'a supernormal dispersion [can] also arise out of connexite or organic connection between the successive terms.

He concludes with Chapter 33, ‘An Outline of a Constructive Theory’. He notes a significant limitation of conventional statistical methods, as then used:

Keynes also deals with the special case where the conventional notion of probability seems reasonable:

His final paragraph reveals Keynes views on the significance of his findings, based on the then conventional view of classical science as traditionally understood at Cambridge:

The above assumptions of non-organic ‘characteristics of atomism and limited variety’ and hence the applicability of the then conventional statistical methods was not long to remain credible, even for the natural sciences, and some economists, notably in the US, applied some of his ideas in the interwar years, although some philosophers continued to find it 'very puzzling indeed'.

Keynes had also noted in Chapter 21 the limitations of 'mathematical expectation' for 'rational' decision making. Keynes developed this point in his more well-known General Theory of Employment, Interest and Money and subsequently, specifically in his thinking on the nature and role of long-term expectation in economics, notably on Animal spirits.

Keynes' ideas found practical application by Turing and Good at Bletchley Park during WWII, which practice formed the basis for the subsequent development of 'modern Bayesian probability', and the notion of imprecise probabilities is now well established in statistics, with a wide range of important applications.

The significance of 'true' uncertainty beyond mere precise probabilities had already been highlighted by Frank Knight and the additional insights of Keynes tended to be overlooked. From the late 60s onwards even this limited aspect began to be less appreciated by economists, and was even disregarded or discounted by many 'Keynesian' economists. After the financial crashes of 2007-9 'mainstream economics' was regarded as having been 'further away' from Keynes' ideas than ever before. But subsequently there was a partial 'return of the master' leading to calls for a 'paradigm shift' building further on Keynes' insights into 'the nature of behaviour under conditions of uncertainty'.

The centenary event organised by the University of Oxford and supported by The Alan Turing Institute for the Treatise and Frank Knight's Risk, Uncertainty, and Profit noted:

However it has often been regarded as more philosophical in nature despite extensive mathematical formulations and its implications for practice.

Informational notes

Citations

Bibliography

• Harrod, R.F. (1951). The life of John Maynard Keynes. London: Macmillan.
• Keynes, John Maynard (1919). The Economic Consequences of the Peace (1 ed.). London: Macmillan & Co., Limited. p. 279. Retrieved 2 June 2016 – via Internet Archive.
• Keynes, John Maynard (1921). A Treatise on Probability. London: Macmillan. Retrieved 2 December 2023.
• Keynes, John Maynard (1922). A revision of the Treaty, being a sequel to The economic consequences of the peace. London: Macmillan. Retrieved 2 December 2023.
• Keynes, John Maynard (1936). General Theory Of Employment, Interest And Money. London: Macmillan.
• Keynes, John Maynard (1949). Two Memoirs: Dr Melchior - a Defeated Enemy and My Early Beliefs. London: Rupert Hart-Davis.
• Skidelsky, Robert (1983). John Maynard Keynes: Hopes Betrayed, 1883–1920. Macmillan.
• Skidelsky, Robert (1992). John Maynard Keynes: the Economist as Saviour 1920-1937. London: Macmillan.
• Skidelsky, Robert (2000). John Maynard Keynes: Fighting for Freedom, 1937–1946. London: Macmillan.
• Skidelsky, Robert (2009). Keynes: Return of the Master. PublicAffairs.

• A Treatise on Probability at Project Gutenberg