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Philosophical Theories of Probability
The Twentieth Century has seen a dramatic rise in the use of probability and statistics in almost all fields of research. This has stimulated many new philosophical ideas on probability.
Philosophical Theories of Probability is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.
Philosophical Theories of Probability is the first book to present a clear, comprehensive and systematic account of these various theories and to explain how they relate to one another. Gillies also offers a distinctive version of the propensity theory of probability, and the intersubjective interpretation, which develops the subjective theory.
- GenresPhilosophyMathematics
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First published January 1, 2000
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Displaying 1 - 8 of 8 reviews
May 21, 2020
I have been reading books on probability for about 5 years, I have also taught a course in decision theory so I was confident this book would just be the same things I had already read before(this is partly why it has been on my list for like 4 years).
Anyway, I was wrong, this book digs into a lot of interesting detail and history that I had never known before. I am writing this review and I am not even done reading the book yet! It is already clear how great this book is. I always want to say that people should not be afraid of a few equations, but though this book is not heavy on math, I do think a lot of the concepts won't make sense without mathematical training so the recommendation must be bounded to those who have taken a stats/prob class.
I understand that he is tempted to put in his own work here, but that is the only limitation of the book. Since his work is relatively new and it is less explored, it is less interesting to explore and just seems like a patchwork on the other theories, though he does try to motivate it by including some quotes from Ramsey and the like.
Anyway, I was wrong, this book digs into a lot of interesting detail and history that I had never known before. I am writing this review and I am not even done reading the book yet! It is already clear how great this book is. I always want to say that people should not be afraid of a few equations, but though this book is not heavy on math, I do think a lot of the concepts won't make sense without mathematical training so the recommendation must be bounded to those who have taken a stats/prob class.
I understand that he is tempted to put in his own work here, but that is the only limitation of the book. Since his work is relatively new and it is less explored, it is less interesting to explore and just seems like a patchwork on the other theories, though he does try to motivate it by including some quotes from Ramsey and the like.
June 20, 2021
Donald Gillies’ Philosophical theories of probability and Timothy Childers’ Philosophy and probability, turn out to be ideal introductions to the field at the level of a beginning graduate student, and complement each other nicely. Gillies focuses on Keynes and Popper and neglects von Mises and Fisher, while Childers covers the latter two. They both supply useful bibliographies.
Why probability is a significant concept: we may ask whether is it scientific at all. Probability theory’s distinguishing feature is its connection between theoria and praxis – unlike what prevails in science but similar to engineering, law and medicine. How so? Most neglect theory in the sense that they merely receive it from others and apply it to their life situation; scientists, on the other hand, tend to go to the opposite extreme (because not motivated by anything other than to know the necessary connection among results in place of any contingent outcome). A scientist, therefore, must be apolitical and disinterested.
Chapter one starts out with a description of the four main types of probability theory, thus Gillies:
1. The logical theory identifies probability with degree of rational belief. It is assumed that given the same evidence, all rational human beings will entertain the same degree of belief in a hypothesis or prediction.
2. The subjective theory identifies probability with the degree of belief of a particular individual. Here it is no longer assumed that all rational human beings with the same evidence will have the same degree of belief in a hypothesis or prediction. Differences in opinion are allowed.
3. The frequency theory defines the probability of an outcome as the limiting frequency with which that outcome appears in a long series of similar events.
4. The propensity theory, or at least one of its versions, takes probability to be a propensity inherent in a set of repeatable conditions. To say that the probability of a particular outcome is p is to claim that the repeatable conditions have a propensity such that, if they were to be repeated a large number of time, they would produce a frequency of the outcome close to p. (p.1)
After saying almost nothing in chapters one and two on historical origins and the classical theory, Gillies turns to logical theory of probability in chapter three, which is illustrative of his empty style. Again remaining descriptive, he merely gives us a patchwork of extracts from works by other authors in place of his own analysis. If the reader be attentive, he will note that all that he can derive from writings such as this is perhaps a definition of key terms (probability as a relation, measurable and non-measurable probabilities, principle of indifference etc.). The presentation, unfortunately, succeeds in being neither coherent nor analytical. The author has nothing in substance original to say, but merely marks up scenarios by others. Ramsey’s critique occupies two pages followed then the Ramsey-de Finetti theorem with proof (involving some mid-ling undergraduate-level computations).
The presentation follows a similar arc in chapters six and seven on Keynes’ propensity theory. Mostly descriptive; few good definitions and proofs, hardly any theorems, no original theme – why would one want to read them? Why not just go to the original literature? At least, one would want some insight so as to compensate for having gone to the effort of reading a book such as this.
Gillies wishes to congratulate himself on having – in accord with prevailing prejudices – framed a pluralist conception of probability according to which there is more than one valid possibly domain-specific interpretation of probability. As illustrated in last chapter where the author distinguishes the sense in which the term probability can be invested with meaning in the natural versus the social sciences. He associates the former with an objective interpretation of probability (principal advocates: von Mises, Fisher, Neyman, Popper) in natural scientific applications and the latter with an epistemological interpretation in applications say to economics (principal advocates: Keynes, Ramsey, de Finetti) and general arguments thereunto. Scarcely all that convincing, in the end. The author’s own original contribution appears to be limited to a five-page closing section on why his pluralist operationalism works in the social but not the natural sciences. Little if anything quantitative, no theorems proved, hardly any advance over common sense and the author’s source, George Soros [Alchemy of Finance: Reading the Mind of the Market, 2nd ed., John Wiley (1994)].
What is the problem here, at root? Our disappointing conclusion is that few probabilists remain with the requisite detachment and daring to aim for systematicity and the industriousness to work out hard paradigmatic solutions in applications. Second – and this is dramatic indeed! – reflection on this work leads the present reviewer to suspect that the conventional distinction between the natural and the social sciences may well be more a matter of presumption than of verified true belief. That is, Popper could be right after all that we live in a quasi-magical world in which elementary propositions such as the law of large numbers are, first-off, empirical questions, and second, actually false (q.v. our review here). To speak from experience, the behavior of probability in the real world is strangely counter to naïve expectation informed by natural science – a stunning discovery!
These unoriginal works by Gillies and Chalmers are nevertheless good enough to merit a serious effort to understand what they really say about the world. The next logical step will be for this reviewer to go to the original literature, starting with Keynes’ treatise of 1921 and Kolmogorov’s from 1933. All his life he has thought himself a Kolmogorovian, but now he suspects himself in fact to be a Keynesian at heart!
Why probability is a significant concept: we may ask whether is it scientific at all. Probability theory’s distinguishing feature is its connection between theoria and praxis – unlike what prevails in science but similar to engineering, law and medicine. How so? Most neglect theory in the sense that they merely receive it from others and apply it to their life situation; scientists, on the other hand, tend to go to the opposite extreme (because not motivated by anything other than to know the necessary connection among results in place of any contingent outcome). A scientist, therefore, must be apolitical and disinterested.
Chapter one starts out with a description of the four main types of probability theory, thus Gillies:
1. The logical theory identifies probability with degree of rational belief. It is assumed that given the same evidence, all rational human beings will entertain the same degree of belief in a hypothesis or prediction.
2. The subjective theory identifies probability with the degree of belief of a particular individual. Here it is no longer assumed that all rational human beings with the same evidence will have the same degree of belief in a hypothesis or prediction. Differences in opinion are allowed.
3. The frequency theory defines the probability of an outcome as the limiting frequency with which that outcome appears in a long series of similar events.
4. The propensity theory, or at least one of its versions, takes probability to be a propensity inherent in a set of repeatable conditions. To say that the probability of a particular outcome is p is to claim that the repeatable conditions have a propensity such that, if they were to be repeated a large number of time, they would produce a frequency of the outcome close to p. (p.1)
After saying almost nothing in chapters one and two on historical origins and the classical theory, Gillies turns to logical theory of probability in chapter three, which is illustrative of his empty style. Again remaining descriptive, he merely gives us a patchwork of extracts from works by other authors in place of his own analysis. If the reader be attentive, he will note that all that he can derive from writings such as this is perhaps a definition of key terms (probability as a relation, measurable and non-measurable probabilities, principle of indifference etc.). The presentation, unfortunately, succeeds in being neither coherent nor analytical. The author has nothing in substance original to say, but merely marks up scenarios by others. Ramsey’s critique occupies two pages followed then the Ramsey-de Finetti theorem with proof (involving some mid-ling undergraduate-level computations).
The presentation follows a similar arc in chapters six and seven on Keynes’ propensity theory. Mostly descriptive; few good definitions and proofs, hardly any theorems, no original theme – why would one want to read them? Why not just go to the original literature? At least, one would want some insight so as to compensate for having gone to the effort of reading a book such as this.
Gillies wishes to congratulate himself on having – in accord with prevailing prejudices – framed a pluralist conception of probability according to which there is more than one valid possibly domain-specific interpretation of probability. As illustrated in last chapter where the author distinguishes the sense in which the term probability can be invested with meaning in the natural versus the social sciences. He associates the former with an objective interpretation of probability (principal advocates: von Mises, Fisher, Neyman, Popper) in natural scientific applications and the latter with an epistemological interpretation in applications say to economics (principal advocates: Keynes, Ramsey, de Finetti) and general arguments thereunto. Scarcely all that convincing, in the end. The author’s own original contribution appears to be limited to a five-page closing section on why his pluralist operationalism works in the social but not the natural sciences. Little if anything quantitative, no theorems proved, hardly any advance over common sense and the author’s source, George Soros [Alchemy of Finance: Reading the Mind of the Market, 2nd ed., John Wiley (1994)].
What is the problem here, at root? Our disappointing conclusion is that few probabilists remain with the requisite detachment and daring to aim for systematicity and the industriousness to work out hard paradigmatic solutions in applications. Second – and this is dramatic indeed! – reflection on this work leads the present reviewer to suspect that the conventional distinction between the natural and the social sciences may well be more a matter of presumption than of verified true belief. That is, Popper could be right after all that we live in a quasi-magical world in which elementary propositions such as the law of large numbers are, first-off, empirical questions, and second, actually false (q.v. our review here). To speak from experience, the behavior of probability in the real world is strangely counter to naïve expectation informed by natural science – a stunning discovery!
These unoriginal works by Gillies and Chalmers are nevertheless good enough to merit a serious effort to understand what they really say about the world. The next logical step will be for this reviewer to go to the original literature, starting with Keynes’ treatise of 1921 and Kolmogorov’s from 1933. All his life he has thought himself a Kolmogorovian, but now he suspects himself in fact to be a Keynesian at heart!
June 12, 2012
Hard to read, hard to understand, so historical, so explained ; for me
August 2, 2017
This is an interesting book for anyone interested in the philosophies behind probability theory. It came as an astonishment to me that there are so many different interpretations of some event A's probability P(A). One studies probability and measure theory for years as a mathematics student, but we never really ask the question what these probabilities actually are - are they limits of observed frequencies are they degrees of belief in some event?
I can recommend this book to anyone interested in philosophy and mathematics!
I can recommend this book to anyone interested in philosophy and mathematics!
December 21, 2020
its contain important concepts but he states in a bad manner
January 27, 2023
Chapters read: The logical theory
January 2, 2016
In this book, Professor Gillies discusses classical probability theory, evaluates the four principle interpretations of probability theory and suggests a hybrid interpretation.
Interestingly enough, classical probability theory, which featured admiration of Newtonian mechanics and a belief in universal determinism, had its start in the use of dice for gambling. I guess the intellectuals who developed this theory needed to know when to hold them and when to throw them. In this chapter, Gillies also addressed another question: Why did the Greek philosophers not address probability theory? There are three possible reasons. The Greeks tended to neglect arithmetic and algebra and did not have the modern Indian/Arabic decimal system. Furthermore, the use of the astrolagus, a small bone in the heel of sheep or deer with four flat sides and two rounded sides, instead of dice made the calculation of empirical probabilities next to impossible.
The current interpretations of probability theory evaluated by Professor Gillies are:
1. The logical theory identifies probability with a degree of rational belief. It assumes that, given the same evidence, all rational humans will hold the same degree of belief in a hypothesis or prediction.
2. The subjective theory identifies probability with the degree of belief of a particular individual and differs from the logical theory in that it allows for differences of opinion.
3. According to the frequency theory, the probability of an outcome is the limiting frequency from which that outcome appears in a long series of similar events.
4. According to the propensity theory, probability is a propensity inherent in a set of repeatable conditions. If the probability of a particular outcome is p, the propensity of the repeatable conditions would produce a frequency of an outcome close to p if repeated a large number of times.
Because there appears to be a polarization between the subjective view, in which probability is the degree of belief of an individual, and the objective view, in which probabilities are features of the material world like charges or masses, Professor Gillies attempts to moderate the difference by suggesting that there are some intermediate cases that he classifies as intersubjective probability. In contrast to subjective theories, which deal with the probability of belief of a particular individual, intersubjective probability acknowledges that individuals are typically parts of a group with some sort of group identity. The probability of a consensus belief is the intersubjective or consensus probability of the social group. In determining the conditions under which a social group will form an intersubjective probability, the following conditions are important:
1. The members of the group must be linked by a common purpose.
2. There must be a flow of information and exchange of ideas between members.
Professor Gillies identifies a spectrum from subjective to objective probability:
1. Subjective: Probabilities represent the degrees of belief of particular individuals.
2. Intersubjective: Probabilities represent the degree of belief of a social group that has reached a consensus.
3. Artefactual: Probabilities can be considered as existing in the material world and objective, but they are the result of interaction between humans and nature. For example, the arrangement of stars is independent of humans, but the grouping of stars into constellations is dependent on the human observer.
4. Objective: Things are fully objective if they exist in the material world quite independently of human beings.
Finally, Professor Gillies concludes that there are three currently viable interpretations of probability: subjective, intersubjective and propensity, corresponding to the three factors in cognition, the individual, the collective and the objective reality.
Interestingly enough, classical probability theory, which featured admiration of Newtonian mechanics and a belief in universal determinism, had its start in the use of dice for gambling. I guess the intellectuals who developed this theory needed to know when to hold them and when to throw them. In this chapter, Gillies also addressed another question: Why did the Greek philosophers not address probability theory? There are three possible reasons. The Greeks tended to neglect arithmetic and algebra and did not have the modern Indian/Arabic decimal system. Furthermore, the use of the astrolagus, a small bone in the heel of sheep or deer with four flat sides and two rounded sides, instead of dice made the calculation of empirical probabilities next to impossible.
The current interpretations of probability theory evaluated by Professor Gillies are:
1. The logical theory identifies probability with a degree of rational belief. It assumes that, given the same evidence, all rational humans will hold the same degree of belief in a hypothesis or prediction.
2. The subjective theory identifies probability with the degree of belief of a particular individual and differs from the logical theory in that it allows for differences of opinion.
3. According to the frequency theory, the probability of an outcome is the limiting frequency from which that outcome appears in a long series of similar events.
4. According to the propensity theory, probability is a propensity inherent in a set of repeatable conditions. If the probability of a particular outcome is p, the propensity of the repeatable conditions would produce a frequency of an outcome close to p if repeated a large number of times.
Because there appears to be a polarization between the subjective view, in which probability is the degree of belief of an individual, and the objective view, in which probabilities are features of the material world like charges or masses, Professor Gillies attempts to moderate the difference by suggesting that there are some intermediate cases that he classifies as intersubjective probability. In contrast to subjective theories, which deal with the probability of belief of a particular individual, intersubjective probability acknowledges that individuals are typically parts of a group with some sort of group identity. The probability of a consensus belief is the intersubjective or consensus probability of the social group. In determining the conditions under which a social group will form an intersubjective probability, the following conditions are important:
1. The members of the group must be linked by a common purpose.
2. There must be a flow of information and exchange of ideas between members.
Professor Gillies identifies a spectrum from subjective to objective probability:
1. Subjective: Probabilities represent the degrees of belief of particular individuals.
2. Intersubjective: Probabilities represent the degree of belief of a social group that has reached a consensus.
3. Artefactual: Probabilities can be considered as existing in the material world and objective, but they are the result of interaction between humans and nature. For example, the arrangement of stars is independent of humans, but the grouping of stars into constellations is dependent on the human observer.
4. Objective: Things are fully objective if they exist in the material world quite independently of human beings.
Finally, Professor Gillies concludes that there are three currently viable interpretations of probability: subjective, intersubjective and propensity, corresponding to the three factors in cognition, the individual, the collective and the objective reality.
November 30, 2014
The interpretation of probabilities has broad impact in many areas of philosophy and science, but is studied by relatively few people who apply probability and statistics. I found the logical, frequentist, subjective, and propensity theories all to capture some important aspect of the overall picture (and all have work to do from their basic formulations to avoid paradox), so I am sympathetic to the multi theory approach. I do think that the Stanford Encylopedia of Philosophy entry is more concise and up to date than this book.
Displaying 1 - 8 of 8 reviews