Ordinal Probability Research Articles

ANALYSIS OF EQUITABILITY IN HUMANITARIAN AID DISTRIBUTION: A CASE STUDY OF NORTH-EAST NIGERIA

This study examines equitability in humanitarian aid distribution in the context of the North-Eastern region of Nigeria, focusing on food security among host communities and internally displaced persons (IDPs). The North-East has been plagued by a decade-long insurgency, resulting in significant displacement of people and dependence on humanitarian assistance. While numerous aid organizations operate in the region, food insecurity and poverty persist at alarming rates, with over 71.5% of the population living below the international poverty line. Using a cross-sectional dataset from the International Organisation for Migration (IOM), this study employs ordinal probability regression and ordinary least squares regression models to analyse the impact of various factors on food security. The result reveals that a higher proportion of IDPs experience high food insecurity compared to host communities, and asset ownership significantly influences food security, with host community households possessing higher asset values. Interestingly, employed households are more likely to be food insecure, suggesting the need for labour market improvements for both groups. These findings underscore the importance of equitable aid distribution, especially in conflict-affected regions, and calls for targeted interventions to address the varying needs of host communities and IDPs. Further research and policy measures are essential to alleviate food insecurity and poverty in the North-Eastern region.

On the Genuine Relevance of the Data-Driven Signal Decomposition-Based Multiscale Permutation Entropy.

Ordinal pattern-based approaches have great potential to capture intrinsic structures of dynamical systems, and therefore, they continue to be developed in various research fields. Among these, the permutation entropy (PE), defined as the Shannon entropy of ordinal probabilities, is an attractive time series complexity measure. Several multiscale variants (MPE) have been proposed in order to bring out hidden structures at different time scales. Multiscaling is achieved by combining linear or nonlinear preprocessing with PE calculation. However, the impact of such a preprocessing on the PE values is not fully characterized. In a previous study, we have theoretically decoupled the contribution of specific signal models to the PE values from that induced by the inner correlations of linear preprocessing filters. A variety of linear filters such as the autoregressive moving average (ARMA), Butterworth, and Chebyshev were tested. The current work is an extension to nonlinear preprocessing and especially to data-driven signal decomposition-based MPE. The empirical mode decomposition, variational mode decomposition, singular spectrum analysis-based decomposition and empirical wavelet transform are considered. We identify possible pitfalls in the interpretation of PE values induced by these nonlinear preprocessing, and hence, we contribute to improving the PE interpretation. The simulated dataset of representative processes such as white Gaussian noise, fractional Gaussian processes, ARMA models and synthetic sEMG signals as well as real-life sEMG signals are tested.

20 years of ordinal patterns: Perspectives and challenges

In 2002, in a seminal article, Bandt and Pompe proposed a new methodology for the analysis of complex time series, now known as Ordinal Analysis. The ordinal methodology is based on the computation of symbols (known as ordinal patters) which are defined in terms of the temporal ordering of data points in a time series, and whose probabilities are known as ordinal probabilities. With the ordinal probabilities the Shannon entropy can be calculated, which is the permutation entropy. Since it was proposed, the ordinal method has found applications in fields as diverse as biomedicine and climatology. However, some properties of ordinal probabilities are still not fully understood, and how to combine the ordinal approach of feature extraction with machine learning techniques for model identification, time series classification or forecasting, remains a challenge. The objective of this perspective article is to present some recent advances and to discuss some open problems.

Regression Modeling of Individual-Patient Correlated Discrete Outcomes with Applications to Cancer Pain Ratings.

Purpose:To formulate and demonstrate methods for regression modeling of probabilities and dispersions for individual-patient longitudinal outcomes taking on discrete numeric values.Methods:Three alternatives for modeling of outcome probabilities are considered. Multinomial probabilities are based on different intercepts and slopes for probabilities of different outcome values. Ordinal probabilities are based on different intercepts and the same slope for probabilities of different outcome values. Censored Poisson probabilities are based on the same intercept and slope for probabilities of different outcome values. Parameters are estimated with extended linear mixed modeling maximizing a likelihood-like function based on the multivariate normal density that accounts for within-patient correlation. Formulas are provided for gradient vectors and Hessian matrices for estimating model parameters. The likelihood-like function is also used to compute cross-validation scores for alternative models and to control an adaptive modeling process for identifying possibly nonlinear functional relationships in predictors for probabilities and dispersions. Example analyses are provided of daily pain ratings for a cancer patient over a period of 97 days.Results:The censored Poisson approach is preferable for modeling these data, and presumably other data sets of this kind, because it generates a competitive model with fewer parameters in less time than the other two approaches. The generated probabilities for this model are distinctly nonlinear in time while the dispersions are distinctly non-constant over time, demonstrating the need for adaptive modeling of such data. The analyses also address the dependence of these daily pain ratings on time and the daily numbers of pain flares. Probabilities and dispersions change differently over time for different numbers of pain flares.Conclusions:Adaptive modeling of daily pain ratings for individual cancer patients is an effective way to identify nonlinear relationships in time as well as in other predictors such as the number of pain flares.

Crash injury severity analysis of E-Bike Riders: A random parameters generalized ordered probit model with heterogeneity in means

Electric bike (E-bike) fatal crashes have increased by 34% during the period between 2014 and 2016, raising a great challenge for traffic safety in China. This study examines the effects of road traffic characteristics, environmental factors, crash characteristics and rider demographic factors on e-bike riders' injury severity. A total of 2222 police-reported crash records of e-bike riders in a representative developing area in China-Hunan province from 2014 to 2016 is used for the current study. To account for the ordinal nature of crash severity and to incorporate unobserved heterogeneity at the observation level, a random parameters generalized ordered probit model with heterogeneity in means (RGOP-HM) is applied for the injury severity analysis. For examining the efficiency of the proposed model in modeling e-bike rider injury severity, ordered probit models, random parameters ordered probit models, random parameters generalized ordered probit models were also estimated. The superiority of RGOP-HM in terms of model fitness statistics indicates the importance of relaxing the limitations of traditional ordinal probability methods. The results of RGOP-HM revealed a wide range of factors associated with the e-bike injuries, including horizontal curves, roads with a high posted speed limit, traffic sign-controlled intersections, dim light, unlighted darkness, single-vehicle crashes, collisions with a heavy motorized vehicle, rider age over 44 (45–59, above 59), and rural areas. Based on the factors contributing to the increased injury severity, several safety implications are proposed from the perspective of engineering, education, and enforcement (3E). This study's findings could provide references for the development of targeted countermeasures to improve e-bike traffic safety in China.

Evaluating Temporal Correlations in Time Series Using Permutation Entropy, Ordinal Probabilities and Machine Learning.

Time series analysis comprises a wide repertoire of methods for extracting information from data sets. Despite great advances in time series analysis, identifying and quantifying the strength of nonlinear temporal correlations remain a challenge. We have recently proposed a new method based on training a machine learning algorithm to predict the temporal correlation parameter, , of flicker noise (FN) time series. The algorithm is trained using as input features the probabilities of ordinal patterns computed from FN time series, , generated with different values of . Then, the ordinal probabilities computed from the time series of interest, , are used as input features to the trained algorithm and that returns a value, , that contains meaningful information about the temporal correlations present in . We have also shown that the difference, , of the permutation entropy (PE) of the time series of interest, , and the PE of a FN time series generated with , , allows the identification of the underlying determinism in . Here, we apply our methodology to different datasets and analyze how and correlate with well-known quantifiers of chaos and complexity. We also discuss the limitations for identifying determinism in highly chaotic time series and in periodic time series contaminated by noise. The open source algorithm is available on Github.

Teaching only about Keynes’s Contributions in Chapter 1 (Plus Possibly Chapters 2, 3, 6 and 26, if Time Permits) of the A Treatise on Probability Falls Far ,far Short of Providing Students with An Informed Foundation about Keynes’s Analysis of Decisionmaking,rationality and expectations

Teaching students about Keynes’s views on decision making ,expectations and rationality requires that the students have had a general introduction and overview of Parts II(interval valued probability), III(finite probabilities)and V(statistics, inexact measurement,and approximation, as discussed by Keynes in Chapter Four of the General Theory on pp.39-40 and 43-44) of Keynes’s A Treatise on Probability (1921) and a detailed coverage of chapters 15 and 17 of his Part II of the A Treatise on Probability on the importance of interval valued probability . Keynes argued in Chapter 15 that decision makers in the real world mainly rely on and use interval valued probability in a heuristic manner. They are not relying on the use of ordinal probability which is simply far, far too weak to provide a foundation for rationality. Meeks’ coverage of chapters 1-4 of Keynes’s A Treatise on Probability plus her possible excursion into chapters 6 and 26 would fail to provide the basic foundation needed by students to understand Keynes’s views in the Treatise on Probability and General Theory. Any primary source of discussion about the links between the A Treatise on Probability and General Theory MUST incorporate Keynes’s exchanges between himself and Hugh Townshend in 1937 and 1938,followed by the conflict between Tinbergen and Keynes in 1938-40 over Keynes’s imprecise probability(the intervals of Part II of the A Treatise on Probability ) using approximation and inexact measurement and Tinbergen’s approach based on precise probability and exact measurement .Meek’s coverage of these sources is simply nonexistent. Most importantly, Meeks needs to avoid the 1973 ,volume 8 version of the A Treatise on Probability in the CWJMK like the plague, as its editorial foreword by R. Braithwaite is completely flawed and erroneous.

Post Keynesian Economics Is Based on Joan Robinson’s Many Canards About Supposed Gaping Holes in Keynes’s Theory: The Real Problem Is Gaping Holes and Gross Ignorance in the Post Keynesian Understanding of Keynes’s a Treatise on Probability

There is no Post Keynesian economist or allied philosopher who can comprehend the following, basic 100 year old fact-Keynes, building on Boole ‘s The Laws of Thought (1854), created an interval valued approach to probability, as well as a decision weight approach, the c coefficient, that re-expresses interval valued probability as non additive and non linear probability, that has nothing to do with radical uncertainty or ordinal probability as asserted continuously for 50 years. Consider the example of R. Skidelsky. R. Skidelsky was very much like Joan Robinson in his academic skills, upon whom he has built his view of Keynes’s contributions. R. Skidelsky, like Joan Robinson, has admitted many times that he is mathematically illiterate, inept, and innumerant. This self admitted fact made Skidelsky very susceptible and receptive to the Frank P. Ramsey myth, recently resurrected by C. MIsak (2020). This myth purports that Ramsey, an 18 year old teenager who came to Cambridge University in 1921, was able to convince Keynes in 1922 that his logical theory of probability, as presented in his 1921 A Treatise on Probability, was full of logical, epistemological and philosophical errors that demolished the entire logical foundation and edifice of Keynes’s theory. Of course, given the fact that Keynes’s theory is built directly on a foundation and edifice of George Boole’s mathematical propositional logic and algebra,which Ramsey (nor any other supporter of Ramsey, such as R.B. Braithwaite) dared not challenge at any time in his lifetime, the belief that Keynes, who had used his approach to approximation and inexact measurement in his Indian Currency and Finance and Economic Consequences of the Peace successfully, would accept Ramsey’s purely academic theory based on additivity and linearity, which Keynes knew did not hold at all in the real world of decision making involving missing evidence and vague, conflicting knowledge, is simply silly. Keynes,of course, realized that Ramsey’s great formal, intellectual skills would make him a great academic thinker. The acceptance by all Post Keynesians and heterodox economists, that Keynes capitulated, either in part or wholly, to Ramsey, has led to a complete failure to properly connect the Keynes of the A Treatise on Probability to the Keynes of the General Theory. This is very similar to the failure of Adam Smith academics to properly connect the Smith of The Theory of Moral Sentiments to the Smith of The Wealth of Nations. Both Keynes’s and Smith’s are, of course, identical in both books. The Ramsey myth has been continually promulgated since 1921. Misak’s 2020 book on Ramsey continues to make assertions about Keynes which are directly contradicted by Keynes himself. The best exposition that explodes the Ramsey myth is the Keynes-Townshend correspondence over the TP of 1937-38, where non numerical probability, weight of the evidence, and the logical theory of probability, and the relation to the GT are covered. There is no mention made of Frank Ramsey or the subjective theory of probability by Keynes or Townshend. The elimination of the Frank Ramsey myth is a necessary step that must be taken before any progress can be made on understanding the linkages between the TP and GT. From Hahn and Skidelsky in 1986 to Walsh in 2001 to Togati in 2020, we can see that the real problem is not any gaps in Keynes’s theory, but the blindness of Post Keynesian and heterodox economists regarding the TP and GT, who are either unable or unwilling to read the TP and/or chapters 20, 21 and appendix to chapter 19 of the GT.

Keynes’s Application of Inexact Measurement and Approximation in Chapter 15 of the A Treatise on Probability Directly Conflicts with R. O’Donnell’s Claims in His Chapter 3 concerning Keynes’s Approach to Measurement in His 1989 Book, 'Keynes, Philosophy, Economics, and Politics'

The claim that Keynes’s non numerical probabilities are ordinal probabilities was shown to be mathematically impossible by Keynes himself in Part II in chapter 15 of the A Treatise on Probability(1921) on pp.160-163 and in chapter 17 on pp.186-194, since Keynes’s non numerical probabilities are identical to Boole’s constituent probabilities. Keynes improved on Boole’s technique and was able to solve Boolean problems much quicker than it took Boole to solve the problems.Part II of the A Treatise on Probability is nearly identical to the analysis provided in his two Cambridge University Fellowships in 1907 and 1908. R. O’Donnell(1989, p.60) attempted to analyze a part of page 160 of the A Treatise on Probability that dealt with Keynes’s inexact measurement and approximation approach using interval probability, but failed to comprehend that the discussion directly contradicts his claims concerning ordinal probability made earlier in his chapter 3 on pp.50-59. His claim that Keynes’s nonnumerical probabilities are ordinal probabilities that can be placed within upper and lower bounds is simply a mathematical error. O’Donnell’s citation of the relevant quote on p.160 of chapter 15 of the A Treatise on Probability leaves out Keynes’s italics emphasis on the word “between”, as well as Keynes’s discussion of the concept of standard probabilities. O’Donnell fails to inform the reader that Keynes provides a detailed mathematical analysis of standard probabilities starting from the last half of page 161 and continuing on pp.162-163.This analysis directly refutes O’Donnell’s claims on pp.50-59 that Keynes’s non numerical probabilities are ordinal. It is even stranger that O’ Donnell recognizes that Keynes’s position was that his method of approximation and inexact measurement was “…frequently adopted in common discourse” in real life.O’Donnell’s admission that inexact measurement and approximation is “ of frequent occurrence in ordinary life”(O’Donnell, p.60) directly negates O’ Donnell’s unsupportable claims about ordinal probability once it is realized that Keynes’s non numerical probabilities are identical to Boole’s constituent probabilities in the mathematical derivation of standard probabilities, which are interval valued probabilities. All of O’Donnell’s results in his 1989 book(his 1982 dissertation) are based on the claim that Keynes’s nonnumerical probabilities are ordinal. An examination of p.60 of his book leads to the strange conclusion that Keynes built his logical theory of probability on ordinal probability while, at the same time, arguing that the use of interval valued probability was a frequent occurrence in ordinary life. None of the major results in O’Donnell’s dissertation are left standing due to his complete and total failure to recognize that Keynes’s non numerical probabilities can’t possibly be ordinal probabilities, given that Keynes’s approach comes directly from George Boole, who not only is not mentioned in O’Donnell’s dissertation (1982) or book(1989), but not mentioned by O’Donnell in any publication in his life.

Keynesian Uncertainty Can Only Be Represented by Imprecise, Non Additive, Interval Valued Probability or Decision Weights Like Keynes’s C: Ordinal Probability Can’t Represent Keynesian Uncertainty

J M Keynes’s two logical relations of rational degree of probability, α, 0≤α≤1 and Evidential Weight of the Argument, w, 0≤w≤1, where w measures the degree of completeness of the evidence, can’t be represented or associated with ordinal probability, although Keynes’s theory of probability can easily deal with ordinal probability with the aid of Keynes’s principle of indifference if symmetries are present. α can be, in some limited instances, represented by a numerical, precise, definite, exact, additive probability if, and only if, w=1, although, in general, for w<1, it must be represented by an non additive interval estimate of probability or by a decision weight, like Keynes’s original, path breaking innovation of his conventional coefficient, c. Nowhere in Boole’s 1854 The Laws of Thought is any concept of ordinal probability discussed analyzed or applied in any detail. This is because ordinal probability can never deal with overlapping estimates of probability, which creates problems of non comparability, non measurability or incommensurability that Boole and Keynes solved with interval valued probability.

A Historical Summary of How a Severe Misinterpretation of the only Diagram in Keynes’s A Treatise on Probability in Chapter III on Page 39 Spread to Philosophers: From G. Meeks (1976) to S.Dow and V.Chick (2012) to S.Bradley(2019)

G. Meeks’s original analysis of the diagram on Page 39 (Page 42 of the CWJMK version in 1973) in chapter III of the A Treatise on Probability in 1976 erred in claiming that Keynes was illustrating ordinal,or rank order, probability measurement. Keynes was actually illustrating interval valued probability, not ordinal probability. Keynes made this very clear in chapter 15 of the A Treatise on Probability in Part II on pp.160-163, as well as in chapters 17, 20, 22, 26, 29, and 30, which all deal with Keynes’s method of inexact measurement and approximation, using lower and upper bounds. Meeks never read Part II or Chapter 15 of the A Treatise on Probability. Meeks’s work was then passed down to R. Skidelsky, A. Carabelli, R. O’Donnell, and many, many other academics, who were attending or were associated with Cambridge University. From this stage, her erroneous work was passed down to S. Dow and V. Chick, and finally to S. Bradley. This erroneous and mistaken view of Keynes’s operational approach to using probability in applications had never appeared in the work of any philosopher until it showed up in April of 2019 in an article published by S. Bradley for the Stanford Encyclopedia of Philosophy.

Keynes’s Major Result From Part II of the A Treatise on Probability Was That, Given That Numerical Probabilities Are Additive, Then Non Numerical Probabilities Must Be Non Additive: Non Additivity Is a Sufficient Condition for Some Degree of Uncertainty to Exist

Keynes’s major accomplishment in Part II of the A Treatise on Probability (1921),which he also accomplished in the 1907 and 1908 Fellowship Dissertation versions, was to show that the addition property of the purely mathematical calculus of probability could only be operational in certain circumstances where the evidential weight of the argument, V(a/h) =degree w, was equal to a w of 1,where 0≤w≤1, so that the decision maker had complete data/evidence set.All relevant information or evidence had be known before any decision had to be made. Thus, all numerical probabilities are additive, so that they would sum to 1. However, there were also non numerical probabilities, which were non additive because of the existence of missing, relevant data or evidence, that would not sum to 1. By far the most important case was sub additive,as opposed to super additive, probabilities, which would sum to less than 1. Note the obvious fact that ordinal probabilities are not only not non additive, since they can’t sum to less than 1, but they also can’t be multiplied. Keynes then showed that non numerical probabilities, which were non additive because they summed to less than 1, had to be interval valued probabilities with a lower (greatest lower bound) and a upper (least upper bound) bound. This, of course, followed directly from pp.265-268 of Boole’s 1854 The Laws of Thought. The adherents of all Heterodox, Post Keynesian, and Institutionalist schools of economics (Skidelsky, O’Donnell, Carabelli, Dow, Lawson, and Runde, for example. See the references) argue that Keynes’s non numerical probabilities were ordinal probabilities. However, this conclusion is an oxymoron, because ordinal probability can have absolutely nothing to do with questions concerned about additivity of probabilities summing to 1 or non additivity of probabilities not summing to one, but being less than 1, since, by definition, ordinal probabilities can’t be summed, added or multiplied. Numerical and ordinal probability can’t deal with non measurability, non comparability or incommensurability. Such problems require nonlinear and non additive approaches to measurement. Only Keynes’s and Boole’s initial ,imprecise approaches to probability, using interval valued probability, can do this. It should be obvious to a mathematically trained reader that Boole and Keynes are the founders of the imprecise probability approach to decision making. Unfortunately, there are no economists or philosophers who covered Part II of the A Treatise on Probability in the 20th or 21st centuries. All readers, like Emile Borel, the French mathematician, skipped Part II of the A Treatise on Probability. This is why the obviously false claim, that Keynes’s non numerical probabilities had to be ordinal probabilities, which directly contradicts Keynes’s position that they are non additive, continues to be generally accepted nearly one hundred years after the publication of Keynes’s A Treatise on Probability.

On the Unfortunate Ignorance of Academics regarding the Technical and Mathematical Tools for Dealing with Uncertainty Developed by J M Keynes in the A Treatise on Probability in 1921 and in His Earlier Fellowship Dissertations in 1907 and 1908: Interval Valued Probability, Decision Weights, and Safety First Analysis

Keynes developed a number of technical, mathematical tools for dealing with the problems of uncertainty (ambiguity, vagueness, indeterminate probabilities, imprecise probabilities) in his A Treatise on Probability in 1921 (and in his earlier Fellowship Dissertations of 1907 and 1908) that continue to be overlooked by practically all academics in the nearly 100 years since its publication in 1921. Keynes’s technical and mathematical modeling developments took place in chapters 15-17 of Part II of the A Treatise on Probability, chapters 20 and 22 of Part III of the A Treatise on Probability, Chapter 26 of Part IV of the A Treatise on Probability, and chapters 29 and 30 of Part V of the A Treatise on Probability. Part II of the A Treatise on Probability included Keynes’s original work dealing with non –additive probabilities that was based on the original work of George Boole in his 1854 The Laws of Thought. Keynes’s Boolean upper and lower bounded probabilities, which Keynes categorized as inexact measurement and approximation, established his complete grasp of the concept of interval valued probability. The American Theodore Hailperin showed in 1986 that both Boole and Keynes were using early versions of linear programming techniques to solve both linear and non linear systems of equations and inequations. Keynes used the name “non –numerical probability“ when discussing his no additive, non linear interval valued probability approach. Keynes’s approach has been erroneously identified as an ordinal probability approach by all Post Keynesians, Institutional and heterodox economists, who were greatly influenced by the 1975 paper on Keynes and uncertainty in HOPE by E R Weintraub, which misinterpreted Keynes in terms of G L S Shackle’s theory of possibility. Part III of the A Treatise on Probability, building on Part II’s interval valued probability concept, developed the concept of finite probability, applicable to both numerical and non numerical (interval valued) probabilities, in order to deal with applications of probability to Keynes’s concepts of induction, based on degrees of analogy, involving degrees of similarity and dissimilarity using human memory ,intuition and pattern recognition. Part IV of the A Treatise on Probability provided the first decision weight approach to decision making, Keynes’s conventional coefficient of weight and risk, c, that incorporated both non linear probability preferences and non additivity by the use of decision weights .These decision weights incorporated Keynes’s logical analysis, contained in chapter 6 of the A Treatise on Probability of his Evidential weight of the argument approach, V(a/h),which was a logical relation like Keynes P(a/h)=α, 0≤α≤1,but which Keynes waited to provide the comparable mathematical analysis for in chapter 26.In chapter 26,Keynes set V(a/h) equal to w,where w was normalized on the unit interval, so that V(a/h) =w, 0≤w≤1, holds. Keynes then defined uncertainty in the General Theory as an inverse function of V in chapter 12 on page 148 in footnote 1. Keynes then defined his conventional coefficient of weight and risk, c, to equal the standard EMV and SEU rules multiplied by the decision weights [(2w/1+w)] for non additivity and [(1/(1+q)] for non linearity. F Y Edgeworth recognized Keynes’s approach as breaking new ground. However, his plea for reader assistance from subscribers to Mind was ignored. Part V of the A Treatise on Probability provided the first “safety first” approach to decision making. As acknowledged by Edgeworth, Keynes used Chebyshev’s Inequality to establish lower bounds for decision problems in order to minimize risk (Keynes, 1921, pp. 353-358). Samuelson, while analyzing Keynes‘s simplified Least Risk analysis on p. 315 of the A Treatise on Probability, unfortunately overlooked Keynes’s generalized approach using Chebyshev’s Inequality in chapter 29. Samuelson also overlooked on this same page Keynes’s development of the logical and mathematical tools needed to specify the investment multiplier in the General Theory, as well as Kahn’s employment multiplier on p.183 of his June, 1931 article in the Economic Journal.

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.

Keynes Demonstrated in Chapter 15 of the A Treatise on Probability That His Non Numerical Probabilities Are Identical to Boole’s Constituent Probabilities: It Is Mathematically Impossible for Keynes’s Non Numerical Probabilities to Be Ordinal Probabilities

The claim that Keynes’s non numerical probabilities are ordinal probabilities was shown to be mathematically impossible by Keynes in chapter 15 of the A Treatise on Probability on pp.162-163 and in chapter 17 on pp.186-194. Keynes’s non numerical probabilities are identical to Boole’s constituent probabilities. Keynes improved on Boole’s technique and was able to solve Boolean problems much quicker than it took Boole to solve the problems. Part II of the A Treatise on Probability is nearly identical to the analysis provided in his two Cambridge University Fellowships in 1907 and 1908.

Keynes’s Logical Theory of Probability Is Concerned with Rational Degrees of Belief Based on Cognition, Which Are Not Mysterious or Mystical, Rank Ordered or Qualitative: F P Ramsey Failed to Grasp Keynes’s Use of Non–Additivity in His Reviews of the a Treatise on Probability

Recent exchanges between two academics on the Internet ,both experts in conventional and unconventional approaches to decision theory, reveal an astounding lack of knowledge and understanding regarding the makeup of Keynes’s logical theory of probability as put forth by Keynes in 1921, approximately 100 years ago. Given this lacuna concerning Keynes, even after nearly 100 years has passed by, there is thus a need to restate some basic fundamental facts about Keynes's approach that can only be attained by reading beyond Part I of the A Treatise on Probability or Chapter III of the A Treatise on Probability. Unfortunately, what academics mean when they say that they have read the A Treatise on Probability is that they have read Part I and/or Chapter III of A Treatise on Probability or Chapter III of the A Treatise on Probability. Parts II, III, IV and V are simply skipped or skimmed over, as was done by F P Ramsey in his reviews of 1922 and 1926. Keynes’s logical theory of probability is based on the upper–lower probability approach of George Boole. Boole is the explicit founder of the logical theory of probability that Keynes built on when writing the TP. Keynes called his approach inexact measurement and approximation. Keynes’s interval valued theory was designed, like Boole’s, to explicitly deal with non measurability, non comparability and incommensurability. Qualitative probability, comparative probability, ordinal probability, or rank ordered probability, just like numerical probability, can’t deal with non measurability, non comparability and incommensurability. The failure to absorb this basic point lies at the heart of all Post Keynesian and Fundamentalist Keynesian interpretations that all ignore George Boole’s fundamental contributions to Keynes’s logical theory of probability.

J M Keynes’s Logical Theory of Probability in the A Treatise on Probability (1921) Is Based on the Interval Valued Approach of Boole and Not on the Principle of Indifference, Which Was of Minor Concern to Keynes, or Ordinal Probability or Rank Ordering

It was George Boole who published the first systematic exposition of a logical theory of probability in 1854 in The Laws of Thought. J M Keynes was the second scholar to publish a systematic, technical exposition of logical probability. This was all carefully presented by the American mathematician,Theodore Hailperin, in 1986. J M Keynes built on Boole using an improved version of Boole’s approach that he probably developed while working with Willian Ernest Johnson, who had also developed an unpublished, systematic approach to logical probability before Keynes. Keynes made very significant improvements to the written materials that made up the subject matter of probability and statistics in the nineteenth century. One of these areas involved Laplace’s earlier Principle of Non-Sufficient Reason. Keynes did systematically improve the nature and range of application of the Principle of Non Sufficient Reason with his Principle of Indifference (POI), which was Keynes’s name for his improved version of Laplace’s Principle of Non Sufficient Reason. However, Keynes NEVER attempted to build a systematic approach operationalizing probability on his POI. Keynes built his own approach on George Boole’s original interval valued approach that used upper and lower probabilities. Keynes called this Approximation or Inexact measurement. The major reason for the failure of economists and philosophers to correctly grasp Keynes’s Inexact, approximation approach to measurement is their failure to comprehend the role played by Boole in Keynes’s A Treatise on Probability. This lacuna is being remedied by Professor R. Arthmar.

Given Keynes's Definitions of Logical Probability and Evidential Weight, It Is Impossible for Keynes’s Approach to Measurement to Be an Ordinal Theory; His 'Non Numerical' Probabilities Must Be Based on Inexact and Imprecise Measurement Using Approximation Involving Boolean Upper and Lower Probability Bounds and Bounded Outcomes

Keynes spent all of his introductory Chapter One of the A Treatise on Probability emphasizing that his logical relation of probability (a relation of similarity – dissimilarity based on analogy) P(a/h)=α, where α came in degrees and 0≤α≤1. In chapter 6, Keynes introduced his logical relation of the evidential weight of the argument, V(a/h), but did not specify what it equaled, although he clearly stated that he would integrate both probability and weight together later. In chapter 26, Keynes specified that V(a/h)= w, where we came in degrees and 0≤w≤1. Keynes showed how to integrate both probability and weight together in his conventional coefficient of weight and risk, c, in chapter 26 of the A Treatise on Probability. Probability and weight were also integrated together by Keynes using the theory of inexact measurement or approximation based on intervals with upper and lower bounds or limits. Ordinal probability can do none of these things. Advocates of the claim that Keynes’s theory was an ordinal theory of probability concentrate exclusively on a misinterpreted diagram that appears near the end of chapter 3 of the A Treatise on Probability. Thus, while Keynes’s theory can easily deal with ordinal probability with the aid of the Principle of Indifference, it is not part of Keynes’s main or major analysis in the A Treatise on Probability, which was interval probability. The same conclusion holds for numerical probability and rational expectations. Rational expectations are a very special case that occurs if w=1. The heterodox, institutionalist, or Post Keynesian approaches, involving irreducible, fundamental, or radical uncertainty, are a very, very special case that occurs only if w=0 in the very long run. Their argument, that measurement is not possible at all, be it exact or inexact, is rejected by Keynes in chapter 4 of the General Theory. Thus, while Keynes’s theory can easily deal with ordinal probability, it is impossible for the ordinal probability to deal with inexact and imprecise interval-valued probability measurement. Keynes’s theory of measurement involves interval-valued probability in the A Treatise on Probability with the numerical and ordinal probability being very special cases of his general theory of interval-valued probability. Keynes’s approach to measurement in the General Theory also involved inexact and imprecise measurement as deployed by Keynes in chapter 4 of the General Theory. No ordinal probability approach to measurement is used by Keynes in the General Theory. The great value of the 1936-1938 Keynes-Townshend exchanges is that they show that Keynes presented an approach to measurement that was imprecise and inexact in both the A Treatise on Probability and the General Theory.

A Demonstration of the Impossibility That Keynes’s Logical Theory of Probability Was Primarily an Ordinal Theory: Keynes’s C Coefficient of the a Treatise on Probability Is a Mathematical Translation of Interval Valued Probability into a Decision Weight Approach That Modifies the Additive, Linear Approach to Mathematical Probability

It is straightforward to demonstrate that Keynes’s logical theory of probability could not primarily be an ordinal theory of probability. The demonstration is based on three main points made by Keynes. Keynes’s first major point in the A Treatise on Probability covered Keynes’s extensive discussion of non additivity (the general inapplicability of the addition rule of the calculus of probabilities) in Part II of the A Treatise on Probability (1921), as well as his many worked out problems using interval valued probability in Part II of the A Treatise on Probability (1921). Any attempt to juxtapose ordinal probability and non additivity is an oxymoron because ordinal probability can’t be added or multiplied. Part II of the A Treatise on Probability has nothing to do with ordinal probability. Keynes’s second major point was that the difficult topic of developing an interval valued approach to probability can be avoided by using decision weights that translate additive probability into non linear and non additive decision weights. Keynes’s conventional coefficient of weight and risk,c, was designed to allow his interval valued probability concept to be translated into non linear ,non additive coefficients or decision weights while starting the analysis using additive probability.It is mathematically and statistically impossible to do this with ordinal probability. Keynes’s third major point in the A Treatise on Probability shows that Keynes’s discussion of statistical frequencies involves his use of interval valued probability through the use of upper and lower limits or bounds,an approach identical to that used by Keynes in Part II. None of Keynes’s results from Parts II ,III, IV, and V of the A Treatise on Probability using either interval valued probability or decision weights, first introduced by Keynes to deal with the problems of non linearity and non additivity are alluded to by any Post Keynesian or Fundamentalist Cambridge Keynesian in any article or book written by them over the last 40 years. O’Donnell (2019) and Dow (2019) are have simply overlooked all of Keynes’s analysis in Part II, IV, and V of the TP dealing with measurement. O’Donnell (2019) and Dow (2019) rely completely on a very severe misinterpretation of pages 38-40 of the TP, as well as the diagram on page 39. The diagram on page 39 of the A Treatise on Probability illustrates the linear and additive probabilities of OAI with the nonlinear (quadratic) and non additive nature of the probabilities of all of the other paths. There is no theory of ordinal probability used, developed, deployed, or applied on pp.38-40 of the TP.

The Impact that Ramsey's Overlooking of Keynes's Non Additive, Interval Valued Probability Approach in His Reviews of the a Treatise on Probability Has Had on Philosophers and Economists

Ramsey’s failure to grasp the interval valued and non additive nature of Keynesian probability assessments in the A Treatise on Probability has led to the misbelief that Keynes’s approach was an ordinal one. While Keynes’s approach easily handles ordinal probability, the theory of the Treatise on Probability is that of “Approximation,” which is interval valued probability. Ramsey’s reviews, however, have been accepted at face value by philosophically interested readers, who fail to grasp that Ramsey has no idea about the fundamental non additive, sub additive, nature of Keynes’s interval valued probability approach. This leads to some very strange assessments of Keynes’s A Treatise on Probability by philosophers who are reading that book.