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Philosophical Theories of Probability (Philosophical Issues in Science)

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This is a wonderful book providing an introduction to the philosophical interpretation of probability.

It is very readable and avoids to a large extend jargon from scientific papers without being to casual. The organization follows a chronological order which allows to follow extensions and critics rather easy. The provided explanations could be sometimes deeps, but they are in general understandable. I guess if more information would be provided it would no longer be such an easy read losing one its attractive features.

I can recommend this book to anyone how wants to get a quick but through overview of this important question.

Classical theory. Equal probabilities are assigned to two events whenever we have no reason to favour one over the other. Thus the probability for each face in a coin flip is 1/2 even if we know that the coin is biased (assuming that we don't know the outcomes of previous flips). This view of probability is epistemological rather than objective. This should not be a surprise since its proponents, notably Laplace, were determinists, which in a sense precludes objective probability altogether.

Logical theory. Keynes' version. Probability = degree of partial entailment = degree of rational belief. Probability is a "Platonic" entity, i.e., not defined in terms of the physical world, which can be perceived through "logical intuition." Probability is relative (to evidence and implicit background knowledge) rather than absolute. The classical principle of indifference is retained in this theory to get it off the ground, especially to determine prior probabilities. But this principle leads to paradoxes. I don't know whether a book I have not seen is red or not, so p(red)=1/2. But is the same way p(blue)=1/2, etc. This problem can be avoided by stipulating that the principle may not be applied to divisible classes (not-red, etc.). But the paradox remains in continuous cases. Suppose we have a mixture of water and wine about which we know only that there is at most 3 times as much of one as the other. Then the principle of indifference gives different probabilities for the identical events wine/water<2 and water/wine>1/2. In fact, the paradox remains to some extent even in finite cases. What is the probability of a head and a tail in two coin flips? If one does not distinguish HT from TH one would say 1/3. But the divisibility criterion forces us to assign 1/4 to each of HH, HT, TH, TT. The problem is that this precludes Bayesian learning: if h is the hypothesis that the next flip will be H, and e is the sequence of flips so far, then p(e|h)=p(e) so by Bayes' theorem p(h|e)=p(h).

Subjective theory. Probability = subjective degrees of belief. This is determined through betting: the subject assigns betting quotients to possible outcomes of an event and is forced to commit to a bet without knowing the stake (which may be negative). The betting quotients are degrees of belief. It is a bit disturbing that this procedure depends on money; for example, one expects people to act differently with respect to huge bets as compared to tiny ones. It has therefore been suggested that the bets should be based on "utility" or "goods" instead of money, but this is obviously hard to make numerically precise. The assumption that betting quotients can be taken as conforming with the probability calculus is reinforced by the Ramsey--De Finetti theorem, which says that a set of betting quotients which violates the probability axioms are incoherent, i.e., Dutch-bookable. "The Ramsey--De Finetti theorem is a remarkable achievement, and clearly demonstrates the superiority of the subjective to the logical theory. Whereas in the logical theory the axioms of probability could only be justified by a vague and unsatisfactory appeal to intuition, in the subjective theory they can be proved rigorously from the eminently plausible condition of coherence. ... In addition, the subjective theory solves the paradoxes of the Principle of Indifference by, in effect, making this principle unnecessary ... In the logical theory, the principle was necessary to obtain the supposedly unique a priori degrees of rational belief, but, according to the subjective theory, there are no unique a priori probabilities. ... There remain, however, some problems connected with the subjective theory, and in particular the question of how probabilities which appear to be objective, such as the probability of a particular isotope of uranium disintegrating in a year, can be explained on this approach." In response "it could be admitted that the examples ... are indeed objective, and consequently that there are at least two different concepts of probability which apply in different circumstances" (Ramsey). Or "it could be claimed that all probabilities are subjective, and that even apparently objective probabilities, such as the ones just described, can be explicated in terms of degree of subjective belief" (De Finetti). This is done by appealing to the washing out of the priors under Bayesian conditionalisation. But washing out occurs only in simplistic situations; in effect, when picking balls from urns. Suppose that we have an urn with a million yellow balls and one black ball; that we draw with repetition; and that we first draw the black ball at the millionth draw. Bayesian conditionalisation will have washed out most of the priors, but only at the cost of committing us to assigning an overwhelming probability to the next ball being yellow. Suppose now that the sun does not rise tomorrow. Will we then say that there is an overwhelming probability that it will rise the day after that? Surely not, because this is not like drawing balls from urns. In such a situation there is no longer any reason to think that priors will wash out. So sometimes people adjust their beliefs in such a way that priors wash out and sometimes they don't. Thus De Finetti's proposed solution of the problem of subjectivity seems backwards and question-begging, in that it identifies as the cause of objectivity a mere symptom of it.

Frequency theory. Von Mises' version. Probability is an empirical science. Start with the phenomena. The object of study is sequences. These satisfy two laws: first, relative frequencies have limits, and second, the progression is random (which Von Mises defines as immune against any betting system, or, equivalently, that limits of relative frequencies are the same for any subsequence). Probabilities are defined as the limits of relative frequencies. So only things that come in sequences can have probabilities, which obviously excludes many common uses of the concept of probability. According to Von Mises, this is a healthy correction of confused pre-scientific usage of the word (cf. the concept of work). De Finetti complained that "If an essential philosophical value is attributed to probability theory, it can only be by assigning to it the task of deepening, explaining or justifying the reasoning by induction. This is not done by Von Mises." Another objection is that since probabilities are limits they are theoretical fictions with no tangible connection to any finite set of data. Von Mises took this to be a harmless idealisation analogous to numerous similar idealisations in physics. De Finetti objected that the analogy is not complete: in physics the idealisation is meant to mediate between an exact theory and imperfect data; in Von Mieses' theory it is the theory itself that is inexact (i.e., says nothing about finite samples).

Propensity theory. Popper introduced the propensity theory by asking: can there be objective probabilities for single events? One motivation was that this seemed to him necessary in quantum mechanics. He first suggested that this could be done by a simple extension of the frequency theory: the probability of a single event is defined as its probability in the sequence in which it occurs. But this clearly has undesirable consequences if the sequence is made up of, e.g., throws with biased and unbiased dice interspersed. To avoid this the concept of sequence can be restricted to sequence of repeated experiments, i.e., sequence characterised by its generating conditions. "Yet, if we look more closely at this apparently slight modification, then we find that it amounts to a transition from the frequency interpretation to the propensity interpretation," for it forces us "to visualise the conditions as endowed with a tendency or disposition, or propensity, to produce sequences whose frequencies are equal to the probabilities; which is precisely what the propensity interpretation asserts" (Popper). Thus the propensity theory gives up Von Mieses' operationalist definition of probability, while maintaining his view that probability theory is about observable random phenomena and that probabilities exist objectively in analogy with, e.g., mass. However, Popper's original emphasis on repeated experiments seems to require subjectivity in the determination of the reference class, i.e., in the determination of which conditions are needed for something to count as a repetition of the *same* experiment. Popper and Miller later avoided this problem by taking propensities to be unique properties determined by the entire state of the universe. But such propensities are metaphysical and untestable. Furthermore, this notion of propensity invites the view that propensity is a generalisation of causation; as Popper put it: "Causation is just a special case of propensity: the case of a propensity equal to 1." But because probabilities have a temporal symmetry which causation lacks, this leads to Humphreys' paradox, e.g.: "Suppose we are given a set of probabilities from which we can deduce that the probability that a certain person died as a result of being shot through the head is 3/4. It would be strange, under these circumstances, to say that this corpse has a propensity of 3/4 to have had its skull perforated by a bullet." The paradox can be avoided by stipulating that event-conditional probabilities are not causal.

Gillies(G)does a very good job covering the subjective,frequency,classical and propensity theories of applied probability.His discussion of his co-authored intersubjective approach,which is to be intermediate between subjective and logical theories of probability ,is interesting but not convincing to this reader.G fails in his exposition of logical theories in general and Keynes's logical theory specifically.G bases his discussion of Keynes's approach on two chapters of the A Treatise on Probability(1921)[TP],chapters 3 and 4 plus the error filled F.Ramsey reviews of 1922 and 1926,as well as Keynes's 1931 4-page review of Ramsey's published works.G thus commits the same error made by Ramsey in his two reviews.Ramsey completely confuses the meaning of "non-numerical probabilities" as well as the meaning of nonmeasurable.Keynes meant for chapter 3 of the TP to be an introduction aimed at interested readers with little or no training in mathematics,logic or statistics.A CAREFUL reading lends no support to Ramsey.On p.34,paragraph 3,Keynes clearly states that by nonmeasurable he means not by "any numerical relation".Thus most probabilities are "non-numerical",which means "not by a single numeral or number".It is in chapters 5,10,15,16 and 17 of the TP that Keynes explains his statements on pages 31-32 of the TP about "numerical limits" and "...since the probability lies between(emphasis added by Keynes)two numerical measures."For Keynes ,ALL probabilities are either precise point estimates(numerical probabilities)or non-numerical interval(set)estimates which are imprecise.Imprecise probabilities are nonmeasurable because they are non-numerical.They are non-numerical because two numbers,not one,are required in order to specify the probability relationship.Consider the two following interval(set)estimates[.4,.6] and [.5,.7].They are Keynesian non-numerical probabilities.They are noncomparable,nonrankable and incommensurable.On pages 135-138 of the TP,Keynes provides a generalized axiomatic analysis of the addition and multiplication rules for probability[conjunction and disjunction]that apply to both numerical and non-numerical probabilities.The axioms operate on sets of propositions.Keynes makes valuable extensions to these axioms in chapters 8 and 15.Unfortunately,neither Ramsey nor Gillies read any of these chapters.All of Gillies'references in this book are to chapters 3 and 4.Given that Keynes builds his entire approach to induction and analogy,in Part 3 of the TP ,on both numerical and non-numerical probabilities,Gillies needs to rewrite those parts of his book dealing with Keynes's logical theory of probability.