Subjective Probability Theory Research Articles

An Examination of the Very Severe Ignorance of Keynes’s A Treatise on Probability Among Heterodox Economists and Their Erroneous Beliefs About Logical and Subjective Probability

Heterodox economists have simply skipped the two most important parts of Keynes’s A Treatise on Probability (1921), Part II and Part V. They basically assess Keynes’s position on probability and uncertainty based on a reading primarily of Chapter III of Part I of the A Treatise on Probability. This results in their failure to grasp Keynes’s inexact measurement – approximation approach to probability in Part II and Keynes’s inexact measurement – approximation approach to statistics in Part V of the TP. Both Part II and V form the basic foundation of Keynes’s approach to logical probability that Keynes built on Boole. Specifically, heterodox economists are ignorant of (i) Keynes’s inexact approach to measurement, based on Boolean approximation that uses lower and upper bounds, when dealing with probability and (ii) Keynes’s inexact approach to measurement ,based on Boolean approximation that uses lower and upper bounds, when dealing with statistics. This results in a belief that Keynes’s method involved an application of ordinal probability only some of the time, because there was supposed to be entities, called non-comparable, non-measureable and incommensurable probabilities, that can’t be analyzed. Heterodox confusions about Keynes’s discussion on pp. 31-36 of the TP concerning unknown probabilities and indeterminate probabilities, where indeterminate probabilities are Boolean in nature and have nothing to do with unknown probabilities, leads heterodox economists into an intellectual quagmire of quicksand that could have been easily avoided if they had covered Parts II and V of A Treatise on Probability. The extensive, but unique, Keynes–Townshend correspondence over the connections between the GT and TP in 1937-38 showed why Keynes’s method of inexact measurement and approximation for both probability and statistics is what links the GT and TP. On questions of probability and expectations, only the TP and the GT are mentioned by Keynes and Townshend in their correspondence. There is no mention of the 1937 QJE article or of fundamental uncertainty or of Frank Ramsey or subjective probability. An examination of both Rosser and Skidelsky reveals that they both simply have no basic understanding about what a logical theory of probability is or what a subjective theory of probability is. Both Rosser and Skidelsky, like Muth before them, confuse the two approaches. An examination of Rosser and Skidelsky illustrates the astounding ignorance of Keynes’s A Treatise on Probability on the part of heterodox economists. Neither Rosser or Skidelsky demonstrate that they have even the slightest understanding of what an interval valued probability is, what Keynes’s method was-inexact measurement and approximation, how uncertainty is related to non additivity, that Keynes was not a subjectivist, that Savage rejected the frequency approach to probability, and that subjective probability distributions can’t possibly converge to an objective probability distribution because objective probability does not exist according to de Finetti.

The Rational Expectations Hypothesis Can Never, Ever Be Shown to Be True (False), Correct (Incorrect), Right (Wrong) or Proven (Not Proven): Ignoring Haavelmo’s Warning about Confusing Deduction (Mathematics, Logic) with Induction (Probability, Statistics, Econometrics) Leads to Oxymorons Such As 'Objectively True Probability Assessments'

Advocates and opponents of the rational expectations hypothesis have both confused statistics with mathematics (logic). Only in mathematics can I prove something to be true, false, correct, incorrect, right or wrong. The best one can do in statistics is, by the use of proper scoring rules, establish that one forecaster (model) is better, more accurate or reliable relative to another forecaster (model). A best forecast can never, ever be a true forecast. Definitions of what the rational expectations hypothesis is are confused, ambiguous, hazy, unclear, and foggy. Practically none of the definitions involve a correct use of statistics and probability. The one technical definition, given by R.Muth in 1961 in an article in Econometrica, is an intellectual mess due to Muth’s own misunderstandings of what statistics and probability meant when the terms subjective theory of probability and objective theory of probability are used. Subjective theories of probability involve degree of belief. Objective theories of probability involve empirical, relative frequencies based on the direct observation of outcomes in a very large number of repeated experiments carried out under identical conditions. An estimate of a subjective probability can never be an estimate of an objective probability. Muth’s definition was that “…for a given information set, the subjective probabilities (distributions) were distributed around a unique, objective probability distribution.” This definition is simply impossible and would be rejected by all of the existing theories of probability (limiting (relative) frequency, subjective, logical, propensity, classical). The Rational expectations hypothesis appears to mean no more than the claim that all producers and consumers, who are rational, make use of all available information so that they all have a complete information set. However, this definition is not substantially different from the very similar definition used by Jeremy Bentham in his disputes with Adam Smith over the nature of probability in 1787 in his revised The Principles of Moral and Legislation. Bentham argued that all men love money at all times, that all men calculate, especially outcomes involving pecuniary concerns, and that all the calculated probabilities were linear, exact, definite and precise,as were all of the outcomes, so that all rational people could calculate their Maximal Utility. Smith, on the other hand, argued that probability was inexact, imprecise and non linear, as risk was not proportional to returns. Keynes’s views on decision making and probability are very similar to Adam Smith’s views.

On R. Muth’s Impossible Dream: Trying to turn Subjective Probabilities into Objective Probabilities

R. Muth’s 1961 Econometrica paper is based on a fundamental and basic confusion on his part about the nature of subjective and objective probability that is impossible to remedy. A bifurcation occurred among scientists, mathematicians, and statisticians, starting in the mid 17th century with the work of Pascal, Fermat,and Cardano, about whether probability was subjective or objective. It could not be both. It had to be one or the other. Subjective probability always dealt with probability as being inside or internal to the human mind. It was a degree of belief of an individual based on his mental, personal, and psychological assessment of the data and factors he believed were involved in determining the behavior of the particular phenomena. On the other hand, Objective probability was always outside or external to the human mind. It was a part of the empirical, real world of phenomena that could be observed and was completely independently from any personal, emotional, mental, psychological or subjective factors. Objective probability was always a relative frequency in occurrence that would approach the true or correct answer in the limit as the number of observations of the particular phenomena approached infinity or some very large number that was an approximation of infinity. Historically, there are no adherents of the subjective approach to probability who accept the objective approach to probability and there are no adherents of the objective approach to probability who accept a subjective approach to probability. There is not, nor could there ever be, a subjective –objective hybrid theory because the definitions of subjective and objective probability directly contradict and conflict with each other. Muth had no historical knowledge of these two completely different definitions, forms, concepts and approaches to probability. Neither did the referees of his paper for Econometrica nor any proponent, advocate, supporter or user of what Muth claimed was the rational expectations hypothesis, which was that, for a given information set, the subjective estimates of probability (subjective probability distributions) were distributed around one single, objective, true, correct objective probability distribution. Muth apparently never realized that his definition was an oxymoron. It is impossible by definition. Nowhere in any of his references are there any that deal with the completely different theories of subjective and objective probability; there are no references to any sources in the history of science, history of probability, history of statistics, philosophy of science, philosophy of probability, or philosophy of statistics. There are no references to methodology or epistemology or to the philosophy of social science. Muth used the 1944 Von Neumann-Morgenstern approach to Expected Utility as the foundation for his theory of consumer and producer maximization of utility and profit in his 1961 article. However, it was made very clear by Von Neumann and Morgenstern that they were going to assume the use of objective frequencies only, because they were only going to deal with risk, where risk was defined to be based on known, objective probability. No subjective probability was allowed. Von Neumann had a completely different theory to deal with (complete) uncertainty when no probabilities were known, which he called min-max. Thus, the individual consumer-producer probability distributions would have to be objective, not subjective, in Muth’s paper in order to be consistent with the theory of risk analyzed by Von Neumann-Morgenstern in 1944. Out of nowhere, Muth starts talking about subjective probabilities (distributions) being distributed, for a given information set, around a true, correct, objective probability distribution. This is impossible and can never occur. There is absolutely no foundation at all in any theory of probability for Muth’s definition of rational expectations. The entire literature on the rational expectations hypothesis needs to be completely and totally reformulated and rewritten so as to be consistent with a foundation in either the subjective theory of probability or the objective theory of probability. Otherwise, rational expectationists need to state that they are not dealing with probability and statistics at all. They are simply assuming the very extreme position of complete knowledge from the Greenspan or Keynesian Continuum or spectrum that ranges from situations of complete ignorance to situations of complete knowledge. Currently, there simply is no foundation in any theory of probability for the rational expectations hypothesis, as defined by Muth in 1961 and accepted in the literature in economics since the Robert Lucas paper of 1972, which simply repeats all of the unsupportable claims made by Muth in his definition of rational expectations in 1961 that confused subjective and objective probability.

Muth’s Rational Expectations Hypothesis, That “…The Subjective Probability Distributions Are Distributed Around an Objective Probability Distribution, for a given Information Set…” is an Oxymoron. It Can Be Corrected by Amending It to Read “…The Subjective Probability Distributions Are Distributed Around a Subjective Consensus Probability Distribution, for a given Information Set…”

Jeremy Bentham’s Maximization of Utility model, set out in chapter three and pages 186-187 of the Principles of Morals and Legislation, where all decision makers knew the exact, precise probabilities and precise, exact consequences of their actions, combined by him with a redefinition of the term to mean ONLY the successful optimization of utility over time. Bentham thus conflated the term rational, which meant planning, preparing, reasoning, thinking and introspection, with the concept of successful utility maximization over time. He also failed to state whether his probabilities were subjective or objective. The maximizing utility approach then combined by him in his later writings with a pendulum-oscillation model. The economy would always self adjust so that the result would be a full employment of all resources equilibrium, so long as the government did not interfere with the private sector economy in any way, was quite, and stayed out of the sunshine of business. Bentham also assumed that there were no external, exogenous shocks from drought, famine, natural disasters, civil war, revolution, or war. However, if such random events took place, their impact would naturally dissipate gradually over time, so that the pendulum, representing the change over time in the macro economy, would always approach the natural equilibrium state of full employment. Bentham incorporated economic growth and technological advance over time into his theory through his concept of forced savings, whereby bankers would loan businessmen the funds that would enable them to bid away resources from the production of consumer goods, in order to produce more advanced investment goods, which would then shift the Production Possibilities Curve outward over time. Bentham’s literary theory is the same as the mathematical rational expectations hypothesis if it is combined with real business cycles and dynamic stochastic general equilibrium theory.The only differences between the two theories is how they were expressed - one in literary form and the other in mathematical form. Muth’s 1961 definition of rational expectations in Econometrica that …the subjective probability distributions are distributed around a true objective probability distribution, for a given information set…, can be viewed as an attempt by him to provide Bentham's model of rational, utility maximizing producers and consumers with a modern mathematical and statistical foundation. However, Muth's definition, that the subjective probabilities (distributions) of such decision makers are distributed around the one true, right, correct objective distribution, for a given information set, is an oxymoron that fails to take into account that there is no subjective probability theory (Bayesian) that accepts the concept of objective probability (Frequentist), while there is no objective probability theory (Frequentist) that accepts the concept of subjective probability (Bayesian). Both the logical and subjective approaches to probability are strident reactions against the frequentist attack on the subjectivity of classical probability and the concepts of initial, a priori probability and inverse probability. Frequentists argue that there is no such thing as an initial or an a priori probability and that inverse probability, in the words of Ronald Fisher, is just wrong. Subjectivists, like Bruno de Finetti, argue that there is no such thing as objective probability. Muth does not cite any sources in his 1961 paper that dealt with the Bayesian (subjective) or Frequentist (objective) distinctions. It is, therefore, impossible to be able to identify what his definitions of subjective and objective probability actually were. It is impossible for subjective probability estimates to be distributed around anything but a consensus subjective probability distribution of experts based on an application of Savage's Principle of Stable Estimation, combined with applications of Proper Scoring Rules to prevent dishonesty in the reporting of claims made about forecasting accuracy vis-a-vis other competing forecasting approaches.

On Paul Davidson’s Critique of Rational Expectations: Aiming at the Wrong Target Using the Wrong Theoretical Construct for the Wrong Reasons

Paul Davidson’s nearly 40 years long critique of Rational Expectations is centered primarily on his claim that the Rational Expectations Hypothesis was built on a foundation that assumed ergodicity. However, the rational expectations hypothesis is not built on the assumption of ergodicity. The concepts of ergodicity and non-ergodicity belong to the Limiting Frequency interpretation of probability. Rational Expectations was supposed to have been built on the Subjective interpretation of probability. Rational Expectations is built on the claim, originally introduced in 1961 in R. Muth’s Econmetrica article, that the subjective probability distributions, for a given information set, are distributed around a true, correct, objective probability distribution. This assumption is incoherent, inconsistent, incomprehensible, and contradictory because there is no existing theory of probability (Classical, Logical, Propensity, Frequency, Subjective) that allows Subjective probabilities (distributions) to become equal to Objective probabilities (distributions). The only restriction allowed by the subjective theory of probability on the preferences of individual decision makers, which is applicable only at the micro level, is that the probabilities, be they initial or updated, are additive and linear. No other restriction is allowed by the Ramsey-Savage-de Finetti subjective Bayesian theory of probability. Rational expectation advocates, proponents, and practitioners add a very large number of additional restrictions on the subjective preferences of decision makers in order to transform subjective probability into an alleged true, correct, right objective probability (distribution). The fatal Achilles’ Heel of Rational Expectations is that there is no extant probability theory that allows subjective probabilities to be transformed into objective probabilities at the macro level or inter temporally throughout time. Savage made in very clear in 1954 in The Foundations of Statistics (p.16) that such an approach was “utterly preposterous” and “complete nonsense” since, as Bruno de Finetti had pointed out, “(objective) probability does not exist”. Paul Davidson’s critique completely overlooks the fundamental error committed in all theories of rational expectations, which is that at some point in time, subjective probabilities become objective probabilities. Nelson Goodman pointed out an additional, fundamental problem for the rational expectations approach of how this change can be expected to occur in the future, which was equivalent to his grue-bleen problems, where at some point in the future, empirically, well established green objects all of a sudden become blue or empirically, well established blue objects all of a sudden become green. Paul Davidson’s critique of rational expectations is completely and fundamentally flawed because Davidson’s “ergodic-nonergodic” duality has absolutely nothing to do with the rational expectations' hypothesis, which is that the subjective probability distributions are distributed around an objective, true distribution for a given information set.

The Overlooked Importance of Benjamin Friedman’s Original, 1979 Critique of the Foundations of Rational Expectations Supplied by Lucas and Sargent: The Fallacy of Long Runism (Conditional a Priorism)

Lucas, Sargent, and all other proponents, advocates, users and supports of the Rational Expectations Hypothesis have overlooked the fact that their claim that decision makers can learn about and apply in the short run knowledge of the long run convergence properties of objective frequency probability distributions, as the number of observations in the series approaches infinity, is problematic. The argument, that they can learn, grasp and apply such long run knowledge and apply it in any short-run period of time, is called the fallacy of long runism (conditional apriorism). Nicholas Rescher demonstrated this repeatedly in work published in the mid to late 1970s. This error was originally made by Charles Sanders Pierce and Hans Reichenbach in their theoretical frequentist approaches to probability. Benjamin Friedman noticed the dependence of the Lucas–Sargent( 1979) claim, that decision-makers can come to know the long run convergence properties of the mechanism that is generating the long run series of observations, knowledge of which can only be learned in the distant, very long run as the limit is approached as the number of observations approaches infinity, about the applicability of the rational expectations hypothesis. Lucas-Sargent sidestepped this criticism by claiming that they could use the Bayesian, Subjective theory of probability as their foundation instead. However, in the early 2000’s, Anderson, Hansen, and Sargent (2003) and Hansen and Sargent (2011) stated that they could not use the subjective theory of probability as the foundation for the Rational Expectations Hypothesis because a large number of restrictions imposed by the rational expectations hypothesis conflicted directly with the theory of subjective probability as developed by L J Savage in 1954. Anderson, Hansen, and Sargent (2003) and Hansen and Sargent (2011) reverted to the claim that the foundation of the Rational Expectations Hypothesis was objective frequentist probability theory. However, they simply repeat the original position published in 1979 by Lucas and Sargent. Benjamin Friedman’s original critique of rational expectations in 1979 is applicable again. The failure of any user, proponent or advocate of the rational expectations hypothesis to demonstrate the forecasting superiority of the rational expectation’s hypothesis over other approaches, through the use of proper scoring rules in the nearly 60 years that have passed since the publication of Muth’s original exposition of the rational expectation’s hypothesis, calls into question the claims made by Lucas and Sargent in 1979 about rational expectations. It is difficult to find a specific theory of probability upon which proponents, advocates, user, or supporters of the rational expectations hypothesis in the literature agree is to be used to support their claims. Currently, the combined use of both subjective and objective theories of probability, which often occurs many times simultaneously in the same literature, fails to recognize that there are no objective theories of probability that recognize any role for subjective probability, and no subjective theories of probability that recognize any role for objective probability. This is most clearly specified by B. de Finetti by his famous saying that “Probability(objective) does not exist.” Rational expectationists have apparently now realized that the subjective theory of probability can’t serve as a foundation for rational expectations. They originally sought to use objective frequentist probability for their foundation until challenged by B. Friedman in 1979. They have now gone back to this foundation but have failed to deal with the objections that were originally leveled in 1979.

Boole, Ramsey and the Keynes-Townshend exchanges on subjective probability

Contrary to what is accepted in many quarters, John M. Keynes actually rejected Frank Ramsey’s subjective theory of probability, in general. He did accept, though, Ramsey’s betting quotient approach, but only in the special case where the weight of the evidence w equaled 1, so that all the probabilities were linear, additive, single number answers. In general, however, Keynes’s probabilities were indeterminate, that is, interval valued with the characteristic of being non-additive and nonlinear because the weight of evidence w was generally less than 1. The Boolean roots of Keynes’s approach to probability and his often-neglected exchanges with Hugh Townshend in 1937-38 provide strong evidence that Keynes never changed his mind on the subjective approach to probability. It is shown that, for Ramsey, the degree of belief is the confidence a decision maker has in the betting odds while, for Keynes, it is the degree of ‘rational’ or logical belief based on George Boole’s logic of propositions.

Given B. De Finetti’s Conclusion that 'Probability (Objective) Does not Exist', Then Rational Expectations Does not Exist Either

Rational Expectations is an approach to probability and expectations that was initiated by Muth in 1961. Muth’s concept is based on an immense muddle and confusion in his mind about subjective and objective concepts of probability that confuses subjective probability with objective probability. Like Nelson Goodman’s Grue concept, where the color green turns into the color blue at some point in the future, at some point rational expectationists claim that subjective probabilities become true, correct, right and objective. This is what they mean by the term rational. Rational is objective and/or true probability. This is impossible and demonstrates a great lack of knowledge about basic theories of probability. There is no existing theory of probability that supports the claims made by rational expectationists. There is no right(wrong), correct, true, or objective probability, probability distribution, or expectation if you assume that you are applying the subjective theory of probability. Likewise, all objective theories of probability deny the existence of subjective probability. This approach has infected the entire economics profession since the early 1970’s and created an even more immense muddle than was originally created by Muth in 1961.

On the Very Severe Logical, Epistemological, Methodological and Philosophical Deficiencies in J. Muth’s Concept of Rational Expectations

J. Muth’s 1961 article on rational expectations was written without any knowledge on Muth’s part about what a subjective theory of probability entails and is based on, or what an objective theory of probability entails and is based on. Muth’s disbelief, that the individual, subjective probability distributions of firms and consumers are distributed around an objective, true, probability distribution, is not supported by any acknowledged scholar who has written on the application of the theory of probability to conduct from either a subjectivist theory of probability perspective (Ramsey, de Finetti, Savage) or an objectivist theory of probability perspective (Venn, R. von Mises, H. Reichenbach, K Popper) in history. Muth’s confusions extend to his analysis of optimizing behavior at the level of the firm and consumer. His choice of exposition used expected utility maximization. Muth implicitly must have been using the Von Neumann-Morgenstern approach under risk, which was based on the application of the relative-limiting frequency interpretation of probability, which is an objective theory of probability. The use of expected utility directly conflicts with Muth’s initial claim that the probability distributions were subjective. Muth needed to have used the Ramsey-de Finetti-Savage Subjective Expected Utility (SEU) approach to support his claims about subjective probability distributions being used by individual consumers and producers. Savage’s principle of stable estimation then shows that, as a sufficient amount of relevant evidence becomes available over time, the individual, different subjective probability distributions will gradually converge toward one subjective, probability distribution. However, there is no such “thing” in the Subjectivist theory of probability as convergence to an objective, true probability distribution or a situation where the subjective probability distributions are distributed around a true, objective, probability distribution. R. Muth’s paper is incomprehensible, as well as being incoherent and contradictory. It is inconsistent with any known theory of objective or subjective probability.

De-Biasing Role Induced Bias Using Bayesian Networks

The merits of using subjective probability theory as a normative standard for evidence evaluation by legal fact-finders have been hotly debated for decades. Critics argue that formal mathematical models only lead to an apparent precision that obfuscates the ad-hoc nature of the many assumptions that underlie the model. Proponents of using subjective probability theory as normative standard for legal decision makers, specifically proponents of using Bayesian networks as decision aids in complex evaluations of evidence, must show that formal models have tangible benefits over the more natural, holistic assessment of evidence by explanatory coherence. This paper demonstrates that the assessment of evidence using a Bayesian network parametrized with values obtained from the decision makers greatly reduces role-induced bias, a bias that has been largely resistant to de-biasing attempts so far. This shows that using Bayesian networks as decision aids can benefit legal decision making.

J. M. Keynes (D. Ellsberg) versus Ramsey (L. J. Savage B. de Finetti): The Role of Interval Valued Probability and the Weight of the Evidence (Ambiguity)

It is mathematically impossible to integrate a concept like Keynes’s weight of the evidence variable, w, or Ellsberg’s rho, ρ, into a subjective theory of probability explicitly because of the definitions of probability given by de Finetti, Ramsey and Savage. The Ramsey –de Finetti–Savage definition of probability identifies the subjective estimate of probability itself to be the degree of confidence or doubt a decision maker has in the outcome. This directly conflicts with Keynes’s view that the degree of confidence can never have anything to do with the logical probability estimate, which is a degree of rational belief. The degree of confidence estimate that a decision maker has in his estimate of logical probability is entirely separate from the probability estimate. They are completely independent from each other as explicitly stated by Keynes in chapter 6 of the A Treatise on Probability. The exact same type of analysis holds in the General Theory. The entire Liquidity Preference theory of the rate of interest erected by Keynes in the General Theory in chapters 13-17 is based on the definition of Liquidity Preference. Liquidity Preference is defined as a function of uncertainty. Uncertainty is then defined as a function of the weight of the evidence. Liquidity Preference is not a function of probability. Therefore, the confidence one has in one’s expectations of future outcomes, when the expectations are based on the estimated probabilities, is a conclusion that makes no sense in the subjective theories of Ramsey, de Finetti, Savage and of de Finetti and Savage. It is impossible to talk about the confidence one has in one’s probability estimate because the subjective probability estimate IS the decision maker’s degree of confidence. Ellsberg’s attempt to convince Savage to change his mind about the nature of and concept of subjective probability, as defined by Savage, was thus doomed to failure from the very beginning. A new definition of subjective probability from Savage (de Finetti or Ramsey) would have been required in order to deal with Ellsberg’s concerns. Savage (de Finetti or Ramsey) would have been forced to abandon their own definition of what subjective probability meant. It was impossible for Savage, de Finetti, Ramsey, or any advocate of the subjective theory of probability, to accept Ellsberg’s rho or Keynes’s w, as a measure of the confidence a decision maker has in his estimate of a probability, without completely changing their definition of what subjective probability means.

Who were the Real Bastard Keynesianss? The Neoclassical Synthesis Keynesians (Samuelson, Hansen, Hicks, Klein, Solow) or the Post Keynesians (Joan Robinson, Austin Robinson, Richard Kahn, G L S Shackle, Sydney Weintraub, Paul Davidson)

J M Keynes developed, taught and applied his IS-LP(LM) model in the time period from December, 1933 to the publication of the General Theory in February, 2 1936. He provided a formal, mathematical analysis of his simultaneous, four equation, IS-LP(LM) model in the General Theory in chapters 15 and 21. Keynes integrated uncertainty, defined as an inverse function of the A Treatise on Probability’s concept of the weight of the evidence, into both the IS and LM curves. Keynes’s uncertainty concept represented different degrees of uncertainty. There was no such thing as radical uncertainty except in the case of investment in fixed, durable capital goods subject to technological innovation, change and obsolescence in the future. Keynes also provided a microeconomic, mathematical foundation for an economic model using elasticity analysis that incorporated his general case and the Classical (Neoclassical) special case in chapters 20 and 21 of the General Theory on pp. 281-286 and pp. 304-306. This formal model, which Keynes called the D-Z model, incorporated changes in expected prices, expected profits and money wages. It served to support the Y income expenditure multiplier model, which was contained in the IS and LP(LM) equations. The neoclassical special case was demonstrated by Keynes to assert that the aggregate supply curve for output as a whole was completely inelastic because consumption and investment goods production was inversely or negatively related because a decrease in consumption would lead to an increase in savings that would lower the rate of interest and increase investment. Keynes’s theory showed that this was not the general case. The general case was demonstrated by Keynes to be one where there was only some degree of elasticity. Keynes incorporated this analysis into his 1936 correspondence with Hicks and Harrod. Neither Hicks nor Harrod ever replied to Keynes’s points concerning his elasticity analysis. The Post Keynesians (Joan Robinson, Austin Robinson, Richard Kahn, G L S Shackle, Sydney Weintraub, Paul Davidson, John Kenneth Galbraith) have denied for over 80 years that Keynes ever had any mathematically correct, developed, formal models of IS-LM or any micro foundations in the General Theory because Keynes supposedly realized that the all-powerful effects of radical (irreducible or fundamental) uncertainty made any such mathematical modeling impossible. This is completely wrong. The Neoclassical Synthesis Keynesians (Samuelson, Hansen, Hicks, Klein, Solow), on the other hand, argued that Keynes did provide a mathematical model of his theory in the General Theory, but that it had to be extracted from a number of various different chapters in the General Theory. They also argued that Keynes’s emphasis on uncertainty, while interesting, was not fundamental to his model as extracted by Meade, Hicks and Harrod, and later by Hansen. Parts of this view are correct. This paper shows that the Post Keynesians are completely wrong while the Neoclassical Synthesis Keynesians have some things right about Keynes’s analysis in the GT and some things wrong about Keynes’s analysis in the GT. What they have wrong, however, can’t be expressed formally in the subjective theory of probability. Tobin would be an example of this in his “liquidity preference as behavior toward risk” paper. Therefore, the real Bastard Keynesians turn out to be the Post Keynesians, who were the first to us the term “Bastard Keynesians”, to attack Samuelson and other such economists.

The Formal, Normative Theories of Savage, De Finetti, and Savage and De Finetti Require Additivity. They Never Introduced Sub or Non Additivity (Interval Valued Probability, Imprecise Probability, Indeterminate Probability) into Their Formal, Normative Theories of Subjective Probability or of Maximizing Subjective Expected Utility

George Boole was the first scholar to present a rigorous, mathematically derived, concept of upper and lower probability with a large number of applications in 1854 in The Laws of Thought. Boole made it clear that this was due to the existence of missing or insufficient information, evidence or knowledge. Boole mentioned a number of times that the problems under consideration would always have incomplete or insufficient data. He termed these kinds of problems as being logically indeterminate. If, however, there was no missing or unavailable information, then the upper and the lower probabilities will become equal to each other. There will no longer be any interval valued, or upper and lower, probabilities. There will be a determinate, exact or precise probability. Additivity will always hold. Keynes introduced his weight of the argument,V,a logical relation, in chapter 6 of the A Treatise on Probability. Its purpose was to grade single pieces of data or evidence. Thus V1(h/x1 x2 x3)>V2(h/ x1 x2) means that the weight of the argument, V1, is greater than V2. The comparable mathematical representation of V was accomplished by Keynes with his weight of the evidence variable, w. w was defined on p.315 of the A Treatise on Probability in chapter 26. W was defined on the unit interval [0,1], just like probability is defined on the unit interval between 0 and 1. A w=1 meant that there was no missing or unavailable evidence or knowledge. The evidence is complete. If w=1, then there can’t be any upper and lower probability estimates. Only if w<1 can interval valued probabilities be defined. If w=0, then no probability can be defined. This is G L S Shackle’s argument. Keynes integrated w into his decision weight formula, c. Keynes called c a “conventional coefficient of weight and risk”. C is defined as being equal to p [1/(1 q)][2w/(1 w)]. One obtains a decision rule when c is multiplied by the outcome A or the utility of the outcome, U(A ), and the adjective ‘Maximize’ is appended. [1/(1 q)] models probability preferences, which are assumed to be linear by Subjectivists. [2w/(1 w)] models weight, w, non linearly. W introduces sub and non additivity. Ellsberg, in a manner similar to Keynes, in 1961 modeled what he defined as the ambiguity of the evidence or data by a variable he defined as rho, ρ, where rho measures the degree of ambiguity. ρ, like Keynes’s w, is defined on the interval [0,1]. A rho equal to 0 means that there is no ambiguity. A rho equal to 1 means that complete ambiguity exists. Interval valued probabilities, based on Ellsberg’s analysis of the work of Smith and Good, is recommended by Ellsberg (Ellsberg,pp.111-125), in cases where 0<ρ<1. Feduzi, Runde and Zappia (2012, 2014, 2017) have claimed repeatedly that de Finetti and Savage formally allowed imprecise and indeterminate probabilities to be used by decision makers in their normative theory of decision making. The only way in which this can be formally incorporated into a decision theory is by the use of a variable similar to Keynes’s w or Ellsberg’s ρ. Nowhere in anything written by de Finetti and/or Savage in their lifetimes does any such variable appear in their formal theoretical analysis or in any of their supporting axioms. Savage stated that he did not know how to integrate such a variable, to take account of vagueness, into the additive probability measure that represented his formal, normative theory. Feduzi, Runde and Zappia (2012, 2014, 2017) have misinterpreted the de Finetti and Savage concerns and sympathy for vacillating, indecisive decision makers, confronted with vague and ambiguous evidence, with their recognition that defining one’s personal probabilities can be difficult, with the erroneous and mistaken belief that de Finetti and Savage incorporated a role for imprecise and indeterminate probabilities within their formal, subjectivist, normative theory of probability and their SEU decision theory that rests completely on additive probability. Feduzi, Runde and Zappia need to cite and show exactly where, in any written article, book, or co-authored paper, that Savage, de Finetti, or Savage and de Finetti introduced sub or non additivity into their formal, normative model. If they are unable to do so, then they need to publicly withdraw their claims in the journals in which they published their claims.

The Foundations of J M Keynes's IS-LM Model in Chapter 15 of the General Theory: Keynes's D-Z Model of Chapters 20 and 21

F. Modigliani presented a special case of Keynes’s General Theory result in 1944 in his “Liquidity Preference and the Theory of Interest and Money”. Modigliani sought to provide the IS-LM model of Hicks’s 1937 Econometrica interpretation of Keynes’s chapter 15 IS-LM model with microeconomic foundations in the theory of the firm that included a production function and labor market. Modigliani overlooked the fact that Keynes had already done exactly that in his chapters 20 and 21 of the General Theory. Section 4 of Keynes’s chapter 15 was the bridge connecting chapter 15 to chapters 20 and 21. Modigliani erred, however, in four ways. First, he used the theory of perfect competition, with its assumptions of perfect information and perfect prediction, and not Keynes’s theory of pure competition. Second, Keynes defined p to be an expected price in the General Theory, whereas Modigliani defined his capital P to be an actual price. This led to his third mistake, which was to define the necessary and sufficient first and second order conditions for optimality, leading to a profit maximum, in the labor market, given decreasing returns, as being where the ACTUAL real wage of labor equaled the marginal productivity of labor. Keynes’ condition is that it is the EXPECTED real wage of labor that equals the marginal productivity of labor. This leads directly to Keynes’s Aggregate Supply Curve of multiple equilibria, which is a locus of the entire set of all possible D-Z intersections, which will lead to one Y value, whereas Modigliani is stuck with only one equilibrium. Modigliani thus has the equivalent of Keynes’s Y-multiplier income expenditure model result from chapter 10 of the General Theory, but no D-Z model of expected prices and expected profits from chapters 20 and 21 of the General Theory. Modigliani’s fourth mistake was that he replaced Keynes’s uncertainty, a function of the weight of the evidence, with risk. This follows from Modigliani’s acceptance of the de Finetti subjective theory of probability, where there is only risk and no uncertainty because all probabilities must be additive, precise probabilities, whereas for Keynes most probabilities must be non-additive, imprecise or indeterminate interval valued probabilities. Modigliani’s paper thus becomes a special case of Keynes’s General Theory analysis in chapters 20 and 21.

On Weckstein's 1959, Keynesian, Weight of the Evidence Based Demolition of G L S Shackle's Attack on the Concept of Probability

R. Weckstein’s 1959 article in the February,1959 Symposium in Metroeconomica on Shackle’s theory of possibility demonstrated that Shackle’s attack on the concept of probability, as an approach to be used by decision makers in the real world, due to the theory of probability’s requirements, such as objectivity, complete ordering, the additivity of objective probabilities, divisibility, repetition, and repeatability, was applicable mainly to the relative and limiting frequency interpretations of probability. Shackle’s attack also held against the subjective theory of probability, which also made the assumptions of complete ordering and additivity. However, Weckstein demonstrated that all of Shackle’s objections to probability per se totally failed when confronted by logical theories of probability, in general, and J M Keynes’s Logical Theory of Probability, in particular, which explicitly criticized classical, limiting frequency, relative frequency, and subjective theories of probability. Keynes explicitly pointed out that all of these other theories of probability were actually special cases, sound and valid in their specific fields of application, but not general in scope or application. Keynes argued that only his logical theory of probability could be a general theory of probability, applicable to single events, infrequent events, and frequent events in the form of Boole’s propositional logic and development of indeterminate/imprecise/determinate probabilities. Specifically, it was the Keynesian concept of the weight of the evidence that leads to the complete refutation of Shackle’s attacks on probability per se. Shackle’s attacks on the concept of probability are sound when made against the Classical, Propensity, Subjective, Relative Frequency, and Limiting Frequency interpretations of probability, but are unsound when confronted with Keynes’s Theory of logical probability. Shackle never responded to Weckstein’s criticisms based on Keynes or logical theories of probability. In fact, he deliberately chose to respond mainly to Weckstein’s mention of Reichenbach’s posit approach to singular events in his article. In fact, Shackle never dealt with Keynes’s logical theory of probability in his lifetime since he never got past page 14 of the A Treatise on Probability. Instead, Shackle attempted to completely bypass the A Treatise on Probability and The General Theory, which is built on Keynes’s weight of the evidence, by claiming that Keynes’s February,1937 article in the Quarterly Journal of Economics represented Keynes’s final view of decision making and probability. A recent article by Derbyshire represents a continuation of Shackle’s severe misrepresentation of probability and especially of Keynes’s work on uncertainty, decision making, and probability. Keynes, building on the logical approach to probability of Boole, had already solved the problems that other theories of probability had in dealing with single, crucial decisions, additivity, and uncertainty. Keynes finished with his analysis in the 1908 Fellowship dissertation, which contained his interval valued, on linear, non-additive approach, when Shackle was five years old. This appears in the TP in Part II. Shackle never read the TP. He read little bits and pieces of it. He then wrote vague and ambiguous comments on the TP that made no sense when he wrote them and which still do not make any sense in 2017. Keynes’s approach would make a better foundation for scenario planning than Shackle’s, with his theory of possibility, if one assumes that academics are capable of actually reading the TP. However, it might be the case that academics are not capable of reading the TP. In that case, then, Shackles’s theory would be a second best choice.

The Unbridgeable Gulf Forever Separating A. Smith and J. M. Keynes from Bentham, Classical Economists, Neoclassical Economists, And Modernn Economists (Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.): The Formal and Mathematical Concepts of Uncertainty, Weight of the Evidence, and Confidence

It is logically impossible for Classical Economists, Neoclassical Economists, and “Modern” Economists (Irving Fisher, Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.) to grasp Smithian and Keynesian economics because both Smith and Keynes explicitly incorporated a specific term, weight of the evidence, into their decision theories to deal with uncertainty besides an additive concept of probability. F. Knight’s distinction between risk and uncertainty is flawed because, although he tried, he failed to explicitly integrate such a term as the “weight of the evidence” into his theory of decision making in order to differentiate uncertainty from risk mathematically and logically. Knight needed an understanding of non-additivity. Integrating such a term into decision theory automatically creates a violation of the purely mathematical laws of the probability calculus because the probabilities can no longer be precise (additive), but must be modeled mathematically as interval valued probabilities (non additive). A concept of weight of evidence thus violates the additivity, complementarity, and linearity requirements needed for the application of either the frequency approach to probability, using objective, precise probabilities or the Subjective, Bayesian approach to probability of Ramsey, de Finetti, Savage, and M Friedman upon which Subjective Expected Utility is build, which requires precise, subjective probabilities. George Boole explicitly provided the first explicit logical and mathematical foundation for the concept of upper and lower probabilities, or interval valued probability, in 1854 with the publication of The Laws of Thought in which he put forth the first explicit logical theory of probability in history. Keynes built on Boole’s foundation and based his logical theory of probability and decision theory on interval valued probability. However, this non-additive and nonlinear approach directly conflicts with the linearity and additivity constructs that underlie the subjective and frequency approaches to probability of I. Fisher, Friedman, Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc. It is not possible for subjectivists to deal with weight because it would automatically make the subjective theory of probability and SEU theories very special theories that could never be more than a limiting case of the work of Adam Smith, George Boole, and J M Keynes. Thus, the intellectual conflict over the role of the weight of the evidence and uncertainty variables in decision theory, which completely divided Smith from Cantillon and Bentham in the eighteenth century, also completely divided Keynes from the Classical and Neoclassical economists of the 20th century. Practically all of the “What did Keynes mean in the General Theory?” literature can be traced back to economists’ gross ignorance of the Keynesian concept of the weight of the evidence and its role in the General Theory. Once Keynes introduced the weight of the evidence variable into the General Theory on p.148, the probabilities(expectations) automatically become interval valued. Non-additivity and non-linearity become the general case, while classical and neoclassical theories based on Benthamite utilitarianism’s use of additive and linear precise probabilities, becomes a very special case of Keynes’s. Keynes’s concept of liquidity preference is directly founded on the logical and mathematical analysis of the weight of the evidence that he incorporated into his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. The conventional coefficient , which substitutes a mathematically more tractable analysis for the much more difficult Boolean interval valued approach, adapted with revisions by Keynes for use in Part II of the A Treatise on Probability, allows a reader to see the explicit role that weight plays in Keynes’s theory .Part II of the A underlies Part III and Part III underlies Part V of the A Treatise on Probability. Liquidity preference automatically brings in interval valued probability if w, the weight of the evidence, is less than 1. Keynes’s analysis of imprecise probability, which occurs in chapter 29 the A Treatise on Probability, is based, as acknowledged by Edgeworth in 1922, on using Chebyshev’s inequality to form a lower bound with the upper bound being given by probability estimates using other known probability distributions, such as the Normal distribution. As more evidence is accumulated, improved estimates will start to approach the upper bound and become precise. Keynes’s analysis of indeterminate probabilities takes place in chapters 3, 5, 10, 15, 16, 17, 20, 22, and 26 of the A Treatise on Probability. “Modern” Economists (Irving Fisher, Friedman Becker, Stigler, Lucas, Sargent, Wallace, Muth, Kydland, Prescott, etc.) can’t deal with Keynes’s general theory of decision making, which underpins the General Theory, because all of their theoretical approaches must use precise probabilities that are additive.

On J M Keynes's Rejection, in General, of Ramsey's Subjective Theory of Probability: The KeynessTownshend Exchanges of 1937 and 1938

J M Keynes rejected Ramsey’s subjective theory of probability in general. He did accept Ramsey’s betting quotient approach in the special case where the weight of the evidence, w, equaled one so that all the probabilities were linear, additive, precise, exact, definite, single number answers. In general, Keynes’s probabilities were indeterminate, interval valued probabilities that were non additive and nonlinear because the weight, w, was less than 1. F.Y. Edgeworth, Bertrand Russell, and Edwin Bidell Wilson all recognized the interval valued nature of Keynes’s probabilities (See my Reviewing the Reviewers of Keynes’s A Treatise on Probability, 2016). The Keynes – Townshend exchanges provide incontrovertible evidence that Keynes never accepted Ramsey’s subjective approach to probability because there was no place in that theory for interval valued probability or for the concept of the weight of the evidence, since, for Ramsey, the subjective estimate of a degree of belief is the confidence a decision maker has in the betting odds while for Keynes, it is the degree of rational belief, not the degree of belief.The Keynes –Townshend exchanges, if carefully read and digested, contains the relevant evidence that allows a reader to conclude that Keynes remained an adherent practitioner of his theory of logical probability his entire life. He never changed his mind.Overlooking the Keynes – Townshend exchanges of 1937-38 explains why economists and academicians have failed to see the close connections that exist between the GT and TP.

Subjective probability theory on ordered normed linear spaces

Subjective probability theory on ordered normed linear spaces

ECONOMIC EVALUATION OF AQUACULTURE INVESTMENTS UNDER CONDITIONS OF RISK AND UNCERTAINTY IN THE MEDITERRANEAN SEA

This study aims to analyze, from a technical and ec onomic point of view, the choice between alternative investments in aquaculture under condit ions of risk and uncertainty. In particular, a comparative analysis has been conducted between an inshore and an offshore farm producing European sea bass in the Mediterranean Sea. The app lied model refers to the theory of subjective probability, assigning a different probability to e xpected incomes during the investment period. Resul ts show an economic convenience of inshore with respect to offshore farming.

Theory and applications of proper scoring rules

A scoring rule $$S(x; q)$$ provides a way of judging the quality of a quoted probability density $$q$$ for a random variable $$X$$ in the light of its outcome $$x$$ . It is called proper if honesty is your best policy, i.e., when you believe $$X$$ has density $$p$$ , your expected score is optimised by the choice $$q = p$$ . The most celebrated proper scoring rule is the logarithmic score, $$S(x; q) = -\log {q(x)}$$ : this is the only proper scoring rule that is local, in the sense of depending on the density function $$q$$ only through its value at the observed value $$x$$ . It is closely connected with likelihood inference, with communication theory, and with minimum description length model selection. However, every statistical decision problem induces a proper scoring rule, so there is a very wide variety of these. Many of them have additional interesting structure and properties. At a theoretical level, any proper scoring rule can be used as a foundational basis for the theory of subjective probability. At an applied level a proper scoring can be used to compare and improve probability forecasts, and, in a parametric setting, as an alternative tool for inference. In this article we give an overview of some uses of proper scoring rules in statistical inference, including frequentist estimation theory and Bayesian model selection with improper priors.