Keynes and probability: An assessment

Abstract

Most interpreters agree that Keynes had a wide-ranging, complex, ‘vision of the world’, which underlies his theoretical contributions. Whenever this is forgotten, as happens in the so-called neoclassical synthesis, not only the original Keynesian spirit goes lost but also, and especially, we lose substantive bricks for our theoretical constructions. The paper considers an important instance of this general rule; namely Keynes's views on the logic of probability, meant as the field concerning human behaviour in an uncertain world (hence connected to, but distinct from, the pure theory of probability, meant as a field of mathematics). The paper begins by recalling the main aspects of the classical and frequentist approaches to probability and the main criticisms they received, pertaining among other things to the limits of their applicability. We then consider Keynes's own views, stressing three aspects: the definition of probability as pertaining to the field of logic, the notion of uncertainty and of the ‘weight of the argument’, the ‘theory of groups’. We then discuss the subjective approach of de Finetti, Ramsey and Savage, and contrast it with Keynes's own views. Finally, we consider the implications of our analysis for the interpretation of Keynes's General Theory, and of his attitude towards econometrics.

Keywords:

• J. M. Keynes
• probability
• risk
• uncertainty

Acknowledgements

A preliminary draft of this paper was presented at a conference organised by the Faculty of Statistics, University of Rome ‘La Sapienza’, on 16 January 2003. Thanks for comments and suggestions are due to Anna Carabelli, Alessandro Cristiani, Marco Dardi, Massimo De Felice, Carlo Panico, Mario Tonveronachi, and to two anonymous referees; the usual caveat applies more strongly than usual.

Notes

1 This latter aspect is stressed by Carabelli: in summarising her interpretation of Keynes's method, she says that for Keynes ‘Probability did not belong to speculative knowledge, but served as a guide of life’ (Carabelli 1988: 234).

2 The Italian mathematician Francesco Paolo Cantelli (1875–1966; quoted by de Finetti 1989: 180) considered the field of probability as composed of different sub-fields, to which different approaches are suited: the ‘urn’ scheme for the cases in which equally probable atomic events can be defined; ‘frequencies’ for fields like insurance; and ‘bets’ for fields such as horse races. de Finetti remarks that the mathematical treatment of the three fields can be analogous – which is true, once probability distributions are assumed as given; what Cantelli's examples stress is that in the different cases probability distributions are obtained in different ways, since the phenomena under consideration differ in nature. There is some similarity between Cantelli's ideas and Gillies's (1973) thesis that an objective interpretation of probability is the appropriate one for classic games of chance.

3 Among other sources, Keynes (1921: 45 ff.) singles out von Kries's Die Principien der Wahrscheimlichkeit of 1886.

4 ‘In order that we may have a rational belief in a proposition p of the degree of certainty, it is necessary that one of two conditions should be fulfilled – (i) that we know p directly; or (ii) that we know a set of propositions h, and also know some secondary propositions q asserting a certainty-relation between p and h’ (Keynes 1921: 17). Here, the second condition applies, with proposition h stating that we are considering a ‘perfect’ die, propositions q providing the logical bridge between the ‘state of the world’ represented by h and the conclusion represented by p, and proposition p stating, for instance, that the possibility of obtaining the same face when twice throwing the same die is one-sixth.

5 Which have the nature of propositions q in the previous footnote.

6 Cf. Costantini (2004: 72–9).

7 The frequentist approach had already been criticised by Edgeworth. Cf. Baccini (2001) and Stigler (1999: 87–128).

8 As Keynes commented (1921: 302), ‘Hume's sceptical criticisms are usually associated with causality; but argument by induction – inference from past particulars to future generalisations – was the real object of his attack. Hume showed, not that inductive methods were false, but that their validity had never been established and that all possible lines of proof seemed equally unpromising.’ Popper's famous example of the white and black swans ([1934]1959: ch. 1, §1) is a variation on this theme. (Although beyond our scope here, we may recall that, while sharing Hume's scepticism about induction, Keynes did not seem to share the Scottish philosopher's scepticism about moral judgement (cf. Carabelli and De Vecchi 1999: 279–80). In any case, let us not lose sight of the fact that scepticism is a far cry from nihilism, as summarised in the motto ‘anything goes’.)

9 ‘It will be convenient to call propositions […] which do not contain assertions about probability-relations, “primary propositions”; and propositions […] which assert the existence of a probability-relation, “secondary propositions”’ (Keynes 1921: 11).

10 ‘If a group does in fact agree on a common betting quotient, we shall call that betting quotient the intersubjective or consensus probability of the social group’ (Gillies and Ietto-Gillies, 1991: 399).

11 In the Treatise on Probability, Keynes seems not to deny such a bi-univocal correspondence, but to make it specific to each individual (due to different individual abilities).

12 ‘What we know and what probability we can attribute to our rational beliefs is, therefore, subjective in the sense of being relative to the individual. But given the body of premisses which our subjective powers and circumstances supply to us, and given the kind of logical relations, upon which arguments can be based and which we have the capacity to perceive, the conclusions, which it is rational for us to draw, stand to these premisses in an objective and wholly logical relation.’ (Keynes 1921: 19). See also the quotation given in footnote 14 below.

13 Carabelli (1988: 99) points to ‘Keynes's concept of rationality as reasonableness or practical rationality’.

14 ‘It is without significance to call a proposition probable unless we specify the knowledge to which we are relating it. To this extent, therefore, probability may be called subjective. But in the sense important to logic, probability is not subjective. It is not, that is to say, subject to human caprice. A proposition is not probable because we think it so. When once the facts are given which determine our knowledge, what is probable or improbable in these circumstances has been fixed objectively, and is independent of our opinion’ (Keynes 1921: 4).

15 On this point see Cristiano (2005); on Moore and Keynes, see Bateman (1988), Carabelli (1988), and O'Donnell (1989). Raffaelli (2006: 162), after recalling Moore's idea ‘that we must fall back on customary rules of conduct’, remarks that in an unpublished paper ‘Keynes blames Moore's conservative conclusion on the implicit acceptance of a wrong theory of probability – the then dominant frequency theory […]. Whereas in Moore's theory probability judgements “can be confirmed or refuted by future events”, Keynes grounds their validity in themselves: “a statement of probability always has reference to the available evidence, and cannot be refuted or confirmed by subsequent events”.’

16 ‘Probability [is here conceived] as a branch of logic’, while ‘in the learned world […] Probability is oftener reckoned with Mathematics than with Logic’ (Keynes 1921: xxv).

17 Keynes insists repeatedly on the fact that it is not always possible to express probabilities as quantitative magnitudes. For instance: ‘There appear to be four alternatives. Either in some cases there is no probability at all; or probabilities do not all belong to a single set of magnitudes measurable in terms of a common unit; or these measures always exist, but in many cases are, and must remain, unknown; or probabilities do belong to such a set and their measures are capable of being determined by us, although we are not always able to determine them in practice’ (Keynes 1921: 33).

18 Knight (1921; esp. 233) distinguishes between risk and uncertainty, the first being the case involving quantitative probabilities and the second the case in which probabilities are unmeasurable. Such a distinction is not to be found in Keynes. Here the distinction would have to keep account of the additional dimension of the ‘weight of the argument’: one might define as risk the case where not only probabilities are measurable, but also the weight of the argument is considerable, so that the agent has – with good reasons – full confidence in his or her own (quantitative) probability statement. Keynes's own definition of risk (Keynes 1921: 348 ff.) is different, indicating the product of the ‘net immediate sacrifice to be made in the hope of obtaining A’ and the ‘probability that this sacrifice will be made in vain’.

19 Keynes (1921: 77). Hence the weight of the argument is a different concept from probability: ‘The weight, to speak metaphorically, measures the sum of the favourable and unfavourable evidence, the probability measures the difference’ (Keynes 1921: 84).

20 Let us recall, however, that Keynes interprets the probability statement as ‘independent of our opinion’. Thus, the weight of the argument could perhaps be correlatively interpreted (although Keynes does not explicitly say so) as determined not by the evidence at the disposal of a single individual agent, but as some sort of common evidence (possibly identifiable with the evidence in principle available to a good pater familias).

21 An anonymous referee suggests that ‘weight and confidence might be considered as correlative terms’. We may add that an increase in the relevant evidence, which according to Keynes always increases the weight of the argument, may decrease the confidence of the subject in a probability statement, whenever the new evidence brings to light the importance of additional, highly uncertain, circumstances hitherto ignored (with a variant of an old adage, ‘only the sage realises how deep is her/his ignorance’). This possibility is suggested by Runde (1990: 283), who provides an in-depth discussion of the notion of the weight of the argument; his interpretation is slightly different from the one presented here, since he does not distinguish between (objective) weight and (subjective) confidence.

22 Keynes (1921: 134 fn.) notes his indebtedness to W. E. Johnson in this connection, but without providing bibliographical references. Johnson (1858–1931), a logician, was Fellow of King's College, Cambridge from 1909.

23 ‘We define a group as containing all the propositions logically involved in any of the premisses or in any conjunction of them; and as excluding all the propositions the contradictories of which are logically included in any of the premisses or in any conjunction of them’ (Keynes 1921: 134). When a is true, then a is true; therefore, although Keynes does not explicitly say this, the group also contains its own premisses. Thus, the first sub-set mentioned above is included in the second sub-set, which coincides with the group.

24 Cf. Roncaglia (1978: 121–4).

25 See the essays collected in de Finetti (1989, 1993), Ramsey (1931) and Savage (1954).

26 The subject must also be assumed to be indifferent towards betting or not betting. Otherwise the ‘supply’ and the ‘demand price’ for a bet will differ by an amount sufficient to compensate the subject for accepting to enter the betting arena. In general, professional gamblers (such as casino owners) require such a compensation.

27 We should recall here, however, de Finetti's (1989: 136) subtle answer: ‘Invece di cercare di riportare tutto all'oggettivo, si può ottenere chiarezza facendo il percorso inverso, riducendo cioè ogni concetto di questo tipo al soggettivo. Il valore di un concetto risulterebbe allora da un'analisi delle ragioni profonde e essenziali che ci hanno spinto, forse inconsciamente, a introdurlo e che ci danno una spiegazione della sua utilità’ (My translation: ‘Rather than trying to relate everything to the objective, we can get clearness by following the opposite route, that is by reducing every concept of this kind to the subjective. The value of a concept would then result from an analysis of the deep and essential reasons which led us, possibly unconsciously, to introduce it and which give us an explanation of its usefulness'.)

28 Thus, Keynes's theory cannot be classified as subjective, though it cannot be classified as objective in a strict definition of the term (as the one given by Gillies and Ietto-Gillies 1991: 394 – quoted above in Section 3).

29 Cf. for instance Costantini (2004: 43, 48, 55). It should be stressed that Bayes's theorem was originally developed in the context of the classical approach to probability. Its association with the subjective approach – as when the latter is labelled as ‘subjective (Bayesian)’– is quite misleading, from the point of view of the history of probability analysis. It is only when the role of our ‘general state of knowledge’ is introduced, that Bayes's theorem can be re-interpreted on logicist or subjective lines.

30 In other terms, the weight of the argument does not appear directly among the variables involved in Bayes's theorem; its role lies in influencing the way in which the new evidence affects the values assigned to prior probabilities, Pr(H/K) and Pr(E/K).

31 Symmetrically, accepting the hypothesis H when Pr(E/H) is very high may also prove a mistake.

32 After recalling ‘the necessity in general of taking into account the a priori probabilities of the different causes’, Keynes (1921: 197) remarks: ‘If a cause is very improbable in itself, the occurrence of an event, which might very easily follow from it, is not necessarily, so long as there are other possible causes, strong evidence in its favour.’

33 Cf. Keynes (1921: 192–205 and 413–17), where Pearson's use of the theorem is criticised. Subjective probability theorists agree on this. Thus de Finetti vehemently criticises ‘those mechanical recipes which indefinitely enrich the statistical recipe books: those recipes which, according to the appropriate definition introduced by Irving J. Good, simply represent “adhockeries”[… namely] decision methods operating as a black box, as a mysterious and unprecise object into which data and questions are introduced at one end and ready-made answers and advice are obtained from the other end’ (de Finetti 1993[1978]: 514).

34 Among the literature on this episode, let us recall Theil (1963), Patinkin (1976), Phelps (1980), Lawson (1985), Pesaran and Smith (1985), Carabelli (1988: 173–93), and Bateman (1990). Lawson (1989) illustrates the background to this debate in the history of the development of econometrics.

35 It may not be entirely irrelevant in this respect, although it is clearly not a decisive point, to recall that Keynes was elected President of the Econometric Society in 1944 and 1945 (Patinkin 1976: 1092). This notwithstanding, Samuelson (1948: 156 fn.) decreed: ‘Keynes’ critical review of Tinbergen's economic business cycle study for the League of Nations reveals that Keynes did not really have the necessary technical knowledge to understand what he was really criticising’.

36 See, for instance, Hendry (1980: 396): ‘Forty years after Keynes wrote, his review [of Tinbergen's 1939 book, Keynes 1939 should still be compulsory reading for all who seek to apply statistical method to economic observations’. See also Pesaran and Smith (1985).

37 This is clearly a reaction to what Keynes perceived as the risk of statistical techniques coming to dominate over economic reasoning; a better metaphor could be that of a cooperative game.

38 As Bateman (1990: 378) remarks, ‘Keynes had a clear interest in induction from the time of his first forays into economics, and […] it influenced the ideas about measuring economic variables, the appropriate application of statistical methods to economic data, and model specification’. Carabelli (1988) and O'Donnell (1989) provide book-length discussions of the subject and many bibliographical references; among the most recent literature, see the essays collected in Runde and Mizuhara (2003).

39 Due to the role of uncertainty, Keynes's notion of equilibrium too is different from the traditional one: ‘Keynes's boot-strap theory of economic equilibrium, dependent only on widespread conventional trust in its stability, was novel’ (Raffaelli 2006: 177).

40 As far as I know, this point – which is related to the first one – has not been made by previous commentators. Gillies and Ietto-Gillies (1991: 404–8) identify ‘intersubjective’ probability (see above, footnote 11) behind Keynes's entrepreneurs’ long-term expectations.

41 On this, see Runde (1994), who also provides a critical illustration of other interpretations (Tobin's, Davidson's, Makowsky's) of the relationship between uncertainty and liquidity preference in Keynes.

42 See the quotation from Keynes (1921: 19) given above in footnote 12.