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Purchase of this book includes free trial access to www.millionbooks.com where you can read more than a million books for free. This is an OCR edition with typos. Excerpt from book: CHAPTER III THE MEASUREMENT OP PROBABILITIES 1. I Have spoken of probability as being concerned with degrees of rational belief. This phrase implies that it is in some sense quantitative and perhaps capable of measurement. The theory of probable arguments must be much occupied, therefore, with comparisons of the respective weights which attach to different arguments. With this question we will now concern ourselves. It has been assumed hitherto as a matter of course that probability is, in the full and literal sense of the word, measurable. I shall have to limit, not extend, the popular doctrine. But. keeping my own theories in the background for the moment, I will begin by discussing some existing opinions on the subject. 2. It has been sometimes supposed that a numerical comparison between the degrees of any pair of probabilities is not only conceivable but is actually within our power. Bentham, for instance, in his Rationale of Judicial Evidence? proposed a scale on which witnesses might mark the degree of their certainty; and others have suggested seriously a ' barometer of probability.' 2 That such comparison is theoretically possible, whether or not we are actually competent in every case to make the comparison, has been the generally accepted opinion. The following quotation 3 puts this point of view very well: " I do not see on what ground it can be doubted that everydefinite state of belief concerning a proposed hypothesis is in itself capable of being represented by a numerical expression, however difficult or impracticable it may be to ascertain its actual value. It would be very difficult to estimate in numbers the vis viva of all of the particles of a human body at any instant; but no one doubts that it is capable of numerical expression. I mention t... Promise Angels is dedicated to bringing you great books at great prices. Whether you read for entertainment, to learn, or for literacy  you will find what you want at promiseangels.com!
Item Specifications...
Studio: www.bnpublishing.com
Pages 466
Est. Packaging Dimensions: Length: 1.25" Width: 5" Height: 8" Weight: 1.16 lbs.
Binding Softcover
Release Date Feb 5, 2008
Publisher www.bnpublishing.com
ISBN 9563100417 ISBN13 9789563100419

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More About John Maynard Keynes


John Maynard Keynes (18831946) is widely considered to have been the most influential economist of the 20th century. His key books include The Economic Consequences of the Peace (1919); A Treatise on Probability (1921); A Tract on Monetary Reform (1923); A Treatise on Money (1930); and his magnum opus, the General Theory of Employment, Interest, and Money (1936). Robert Skidelsky is Emeritus Professor of Political Economy at the University of Warwick, England, and a member of the House of Lords. His threevolume biography of Keynes received numerous awards, including the Lionel Gelber Prize and the Council on Foreign Relations Prize.
John Maynard Keynes was born in 1883 and died in 1946 and has an academic affiliation as follows  University of Cambridge.
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 Oustanding! Feb 14, 2008 
This is an outstanding book from an outstanding mind better known from his revolutionary economic ideas.    Contains the first mathematically designed interval estimate approach to probability in history Aug 11, 2007 
In this path breaking contribution to the logic of probability,Keynes showed how to adapt the work of George Boole for the purpose of estimating probabilities.Keynes is the first scholar in history to explicitly emphasize the importance of interval estimates in decision making.For Keynes there are only two types of probability estimates,point estimates and interval estimates;ordinal ranking can be incorporated into the interval estimate classification.Unfortunately,Keynes decided to call interval estimates " nonnumerical " probabilities.His reasoning is really quite obvious.A precise estimate of probability used a single numeral for the point estimate.Therefore,an imprecise,indeterminate estimate of probability used two numerals to denote an interval(set).Thus, an interval estimate is not based on the use of a single number or numeral but two numbers or numerals.This is what Ramsey fundamentally objected to. These types of probabilities are ,thus ," nonnumerical " because you are not using a single numeral.In 1922 and 1926,Frank Ramsey reviewed Keynes's book based on his very brief,partial, and haphazard reading of parts of chapters 14 plus 3 pages from Part II and 4 pages from Part V.Keynes's discussion of nonnumerical probabilities takes place in detail in chapters 5,10,15 and 17,although there is a clear discussion of intervals for the careful reader contained in chapter 3.Keynes then applies his new approach to induction and analogy in chapters 20 and 22,using his concept of " finite probability " ,which applies to both precise ,numerical probabilities and imprecise, nonnumerical probabilities.All of Keynes's discoveries ,however,were ignored by the ignorant Ramsey.It is unfortunate that the editorial foreword to the 1973 Collected Writings of JMK edition of the TP, written by Richard Braithwaite ,simply repeats all of the errors made by Ramsey in his reviews.Consider Braithwaite's paraphrase of Ramsey's argument that " On Keynes's theory it is something of a mystery why the probability relations should be governed by the probability calculus."(p.xx,1973).The answer is really quite simple. First,the " non numerical " interval estimates will not be governed by the probability calculus rules of conjunction and disjunction.Second,numerical probability calculations,such as the bluegreen taxi cab problem of Tversky and Kahneman,will only satisfy the probability calculus if the weight of the evidence,w,is equal to 1,where w is defined as an element on the unit interval between 0 and 1 and measures the relative completeness of the available evidence upon which the probability estimates are to be calculatedKeynes was the first scholar in history to define and create an index to measure the weight of the evidence .[To this day(2007)one can regularly read about Keynes's " strange,mysterious,unfathomable,undefined " nonnumerical probabilities in literally hundreds of economics and philosophy journal articles and books that have been written about Keynes's approach to probability since the Ramsey reviews were first published 80 years ago.Ramsey's reviews are still cited as " overwhelming " evidence that Keynes agreed that Ramsey's critique had completely demolished the entire structure of his logical approach to probability.Nothing could be further from the truth.Ramsey's theory is a special case that holds only when all the probabilities are numerical,additive,precise, and unique.This requires that the weight of the evidence ,w, be equal to 1 so that the probability calculus(addition and multiplication rules) of mathematical probability can be applied.Ramsey's reviews were so poor that Keynes and Bertrand Russell attempted to downplay their relevance so as to save Ramsey from being embarrassed in the academic community.]Keynes then showed that interval estimates,because they frequently overlap and/or will be contained inside another interval,would very likely also,in many cases,be nonmeasurable,noncomparable and/or nonrankable if a decision maker used such order preserving operators like " greater than or equal to " or "less than or equal to " or "equal to".While this is quite obvious to any reader of Part II of the TP who has covered pp.160163 and pp.186194 of the TP or the comparable pages from Booles 1854 classic,p.268,pp.276278,pp.281283,etc.,it went completely over Ramsey's head.Ramsey had never comprehended what it was that Booole was doing in Part V of the Laws of Thought. Keynes's second major advance was to create his "conventional coefficient of weight and risk ", c=p/(1+q)[2w/(1+w)] in sections 7 and 8 of chapter 26 . The goal of the decision maker is to Maximize cA,where A is some outcome.This decision rule solves most of the paradoxes and anomalies that plague subjective expected utility theory.A major accomplishment made by Keynes in chapter 26 of the TP was to specify that the weight of the evidence variable,w,was defined on the unit interval [0,1].It would be forty years before Daniel Ellsberg would define his practically identical variable,rho,on the unit interval between 0 and 1 also,where rho measured the degree of confidence in the decision maker's information base.Since these two measures are one to one onto and isomorphic,Keynesian weight(uncertainty in the General Theory) and Ellsbergian ambiguity measure the same thing and are interchangeable.This means that Ellsberg's analysis can be applied when studying the GT and used to buttress Keynes's theory of liquidity preference in the GT.You simply substitute rho for w in the c formula above or substitute w for rho in Ellsberg's decision rules.In Part 5 of this book ,Keynes showed how one could use Chebyshev's Inequality as a lower bound to the normal probability distributions overly precise and inaccurate point estimate . Part 5 of the Treatise,which is based on Part III's analysis of induction and analogy, also includes Keynes's advocacy of the Lexis Q test for the dynamic stability of a statistical frequency[law of large numbers]over time.It is this part of the TP that forms the basis,along with chapter 17's analysis of the misuse of the Normal distribution in science and social science,of Keynes's exchange with Tinbergen over the logical foundations of econometrics(the use of multiple regression and correlation analysis in the study of time series data) in the Economic Journal in 19391940.Keynes pointed out that ,in order to justify his assumption of normality,Tinbergen needed to apply the Lexis Q test to his time series data.Tinbergen never applied either that test or the Chi Square test for goodness of fit.TINBERGEN NEVER APPLIED ANY GOODNESS OF FIT TEST TO HIS TIME SERIES DATA IN HIS LIFETIME.This will then bring the reader back to Keynes's chapter 8 of the Treatise ,where he presents his own logical frequency interpretation of probability as a special case of his general logical approach to probability.This chapter includes his criticism of Venn's particular version of a frequency approach.
   Treats mathematical probability as a limiting case of logical probability Sep 18, 2006 
In this path breaking contribution to the logic of probability,Keynes showed how to adapt the work of George Boole for the purpose of estimating probabilities.Keynes is the first scholar in history to explicitly emphasize the importance of interval estimates in decision making.For Keynes there are only two types of probability estimates,point estimates and interval estimates.Unfortunately,Keynes decided to call interval estimates " nonnumerical "probabilities.His reasoning is really quite obvious.A precise estimate of probability used a single numeral for the point estimate.Therefore,an imprecise estimate of probability used two numerals to denote an interval(set).Thus, an interval estimate is not based on a single numeral but two. These types of probabilities are thus " nonnumerical "because you are not using a single numeral.In 1922 and 1926,Frank Ramsey reviewed Keynes's book based on his reading of chapters 14 plus 3 pages from Part II and 4 pages from Part V.Keynes's discussion of nonnumerical probabilities takes place in chapters 5,10,15 and 17.Keynes then applies his new approach to induction and analogy in chapters 20 and 22,using his concept of " finite probability " ,which applies to both precise ,numerical probabilities and imprecise, nonnumerical probabilities.All of Keynes's discoveries ,however,were ignored by the ignorant Ramsey.It is unfortunate that the editorial foreword to the 1973 Collected Writings of JMK edition of the TP, written by Richard Braithwaite ,simply repeats all of the errors made by Ramsey in his reviews.Consider Braithwaite's paraphrase of Ramsey's argument that " On Keynes's theory it is something of a mystery why the probability relations should be governed by the probability calculus."(p.xx,1973).The answer is quite simple. First,the " non numerical " interval estimates will not be governed by the probability calculus.Second,numerical probability calculations,such as the bluegreen taxi cab problem of Tversky and Kahneman,will only satisfy the probability calculus if the weight of the evidence,w,is equal to 1,where w is defined as an element on the unit interval between 0 and 1 and measures the relative completeness of the available evidence upon which the probability estimates are to be calculated.(To this day(2006)one can regularly read about Keynes's " strange,mysterious,unfathomable,undefined " nonnumerical probabilities in literally hundreds of economics and philosophy journal articles and books that have been written about Keynes's approach to probability since the Ramsey reviews were first published 80 years ago.These reviews are still cited as " overwhelming " evidence that Keynes agreed that Ramsey's critique had completely demolished the entire structure of his logical approach to probability.Nothing could be further from the truth.Ramsey's reviews were so poor that Keynes and Bertrand Russell attempted to downplay their relevance so as to save Ramsey from being embarrassed in the academic community.)Keynes then showed that interval estimates,because they frequently overlap,would very likely also,in many cases,be nonmeasurable,noncomparable and/or nonrankable if a decision maker used such order preserving operators like " greater than or equal to " or "less than or equal to ".While this is quite obvious to any reader of Part II of the TP,it went completely over Ramsey's head. Keynes's second major advance was to create his "conventional coefficient of weight and risk", c=p/(1+q)[2w/(1+w)] in sections 7 and 8 of chapter 26 . The goal of the decision maker is to Maximize cA,where A is some outcome.This decision rule solves most of the paradoxes and anomalies that plague subjective expected utility theory.A major accomplishment made by Keynes in chapter 26 of the TP was to specify that the weight of the evidence variable,w,was defined on the unit interval [0,1].It would be forty years before Daniel Ellsberg would define his practically identical variable,rho,on the unit interval between 0 and 1 also,where rho measured the degree of confidence in the decision maker's information base.Since these two measures are one to one onto and isomorphic,Keynesian weight(uncertainty in the General Theory) and Ellsbergian ambiguity measure the same thing and are interchangeable.This means that Ellsberg's analysis can be applied when studying the GT and used to buttress Keynes's theory of liquidity preference in the GT.In Part 5 of this book ,Keynes showed how one could use Chebyshev's Inequality as a lower bound to the normal probability distributions overly precise and inaccurate point estimate . Part 5 of the Treatise also includes Keynes's advocacy of the Lexis Q test for stability of a statistical frequency[law of large numbers].It is this part of the TP that forms the basis,along with chapter 17,of Keynes's exchange with Tinbergen over the logical foundations of econometrics in the Economic Journal in 19391940.Keynes pointed out that ,in order to justify his assumption of normality,Tinbergen needed to apply the Lexis Q test.Tinbergen never applied either that test or the Chi Square test for goodness of fit.This will then bring the reader back to Keynes's chapter 8 of the Treatise ,where he presents his own logical frequency interpretation of probability as a special case of his general logical approach to probability after criticizing Venn's particular version of a frequency approach.
   The best book published on the foundations of probability May 17, 2005 
In this path breaking contribution to the logic of probability,Keynes showed how to adapt the work of George Boole for the purpose of estimating probabilities.Keynes is the first scholar in history to explicitly emphasize the importance of interval estimates in decision making.For Keynes there are only two types of probability estimates,point estimates and interval estimates.Unfortunately,Keynes decided to call interval estimates "nonnumerical"probabilities.His reasoning is really quite obvious.A precise estimate of probability used a single numeral for the point estimate.Therefore,an imprecise estimate of probability used two numerals to denote an interval(set).Thus, an interval estimate is not based on a single numeral but two. These types of probabilities are thus "nonnumerical"because you are not using a single numeral.In 1922 and 1926,Frank Ramsey reviewed Keynes's book based on his reading of chapters 14 plus 3 pages from Part two and 4 pages from Part five.Keynes's discussion of nonnumerical probabilities takes place in chapters 5,10,15 and 17.Keynes then applies his new approach to induction and analogy in chapters 20 and 22,using his concept of "finite probability"which applies to both precise numerical probabilities and imprecise nonnumerical probabilities.All of Keynes's discoveries ,however,were ignored by the ignorant Ramsey.(To this day(2005)one can regularly read about Keynes's "strange,mysterious,unfathomable,undefined"nonnumerical probabilities in literally hundreds of economics and philosophy journal articlesthat are based primarily on Ramsey's reviews.These reviews are still cited as "overwhelming" evidence that Keynes agreed that Ramsey's critique had demolished the entire structure of his logical approach to probability.Nothing could be further from the truth.Ramsey's reviews were so poor that Keynes and Bertrand Russell attempted to downplay their relevance so as to save Ramsey from being embarrassed in the academic community.)Keynes then showed that interval estimates,because they overlap,would very likely also,in many cases,be noncomparable and/or nonrankable if a decision maker used such order preserving operators like"greater than or equal to"or "less than or equal to".While this is quite obvious,it went completely over Ramsey's head. Keynes's second major advance was to create his "conventional coefficient of weight and risk", c=p/(1+q)[2w/(1+w)]. The goal of the decision maker is to Maximize cA,where A is some outcome.This decision rule solves all of the paradoxes and anomalies that plague subjective expected utility theory.A major accomplishment made by Keynes in chapter 26 of the TP was to specify that the weight of the evidence variable,w,was defined on the unit interval [0,1].It would be forty years before Daniel Ellsberg would define his practically identical variable,rho,on the unit interval between 0 and 1 also,where rho measured the degree of confidence in the decision maker's information base.Since these two measures are one to one onto and isomorphic,Keynesian weight(uncertainty in the General Theory) and Ellsbergian ambiguity measure the same thing and are interchangeable.This means that Ellsberg's analysis can be applied when studying the GT and used to buttress Keynes's theory of liquidity preference in the GT.In Part 5 of this book ,Keynes showed how one could use Chebyshev's Inequality as a lower bound to the normal probability distributions overly precise point estimate . Part 5 of the Treatise also includes Keynes's advocacy of the Lexis Q test for stability of a statistical frequency[law of large numbers].It is this part of the TP that forms the basis,along with chapter 17,of Keynes's exchange with Tinbergen over the logical foundations of econometrics in the Economic Journal in 19391940.Keynes pointed out that ,in order to justify his assumption of normality,Tinbergen needed to apply the Lexis Q test.Tinbergen never applied either that test or the Chi Square test for goodness of fit.This will then bring the reader back to Keynes's chapter 8 of the Treatise ,where he presents his own logical frequency interpretation of probability as a special case of his general logical approach to probability.
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