Conformity to the full probability calculus thus seems to be necessary and sufficient for coherence. Note, however, that de Finetti—the arch subjectivist and proponent of the Dutch Book argument—was an opponent of countable additivity e. But let us return to the betting analysis of credences. The betting analysis gives an operational definition of subjective probability, and indeed it inherits some of the difficulties of operationalism in general, and of behaviorism in particular.
Moreover, as Ramsey points out, placing the very bet may alter your state of opinion. Trivially, it does so regarding matters involving the bet itself e. Less trivially, placing the bet may change the world, and hence your opinions, in other ways. And then the bet may concern an event such that, were it to occur, you would no longer value the pay-off the same way. During the August 11, solar eclipse in the UK, a man placed a bet that would have paid a million pounds if the world came to an end.
The problems may be avoided by identifying your degree of belief in a proposition with the betting price you regard as fair, whether or not you enter into such a bet; it corresponds to the betting odds that you believe confer no advantage or disadvantage to either side of the bet Howson and Urbach At your fair price, you should be indifferent between taking either side. For example, a sum that can be divided into only parts will leave probability measurements imprecise beyond the second decimal place, conflating probabilities that should be distinguished e.
More significantly, if utility is not a linear function of such sums, then the size of the prize will make a difference to the putative probability: winning a dollar means more to a pauper more than it does to Bill Gates, and this may be reflected in their betting behaviors in ways that have nothing to do with their genuine probability assignments. De Finetti responds to this problem by suggesting that the prizes be kept small; that, however, only creates the opposite problem that agents may be reluctant to bother about trifles, as Ramsey points out.
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Better, then, to let the prizes be measured in utilities: after all, utility is infinitely divisible, and utility is a linear function of utility. After all, there is a sense in which every decision is a bet, as Ramsey observed. Utilities desirabilities of outcomes, their probabilities, and rational preferences are all intimately linked. And most remarkably, Ramsey and later, Savage and Jeffrey derives both probabilities and utilities from rational preferences alone. The result of a coin toss is typically like this for most of us.
He is then able to define equality of differences in utility for any outcomes over which the agent has preferences. It turns out that ratios of utility-differences are invariant — the same whichever representative utility function we choose. This fact allows Ramsey to define degrees of belief as ratios of such differences.
Ramsey shows that degrees of belief so derived obey the probability calculus with finite additivity. For a given set of such preferences, he generates a class of utility functions, each a positive linear transformation of the other i.
See Buchak for more discussion. Some of the difficulties with the behavioristic betting analysis of degrees of belief can now be resolved by moving to an analysis of degrees of belief that is functionalist in spirit. There is a deep issue that underlies all of these accounts of subjective probability. They all presuppose the existence of necessary connections between desire-like states and belief-like states, rendered explicit in the connections between preferences and probabilities. In response, one might insist that such connections are at best contingent, and indeed can be imagined to be absent.
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Think of an idealized Zen Buddhist monk, devoid of any preferences, who dispassionately surveys the world before him, forming beliefs but no desires. It could be replied that such an agent is not so easily imagined after all — even if the monk does not value worldly goods, he will still prefer some things to others e. Once desires enter the picture, they may also have unwanted consequences. The derivation of them from preferences makes them ascertainable to the extent that his or her preferences are known. The expected utility representation makes it virtually analytic that an agent should be guided by probabilities — after all, the probabilities are her own, and they are fed into the formula for expected utility in order to determine what it is rational for her to do.
So the applicability to rational decision criterion is clearly met. But do they function as a good guide? Here it is useful to distinguish different versions of subjectivism. Orthodox Bayesians in the style of de Finetti recognize no rational constraints on subjective probabilities beyond:.
This is a permissive epistemology, licensing doxastic states that we would normally call crazy.
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Thus, you could assign probability 1 to this sentence ruling the universe, while upholding such extreme subjectivism. Some subjectivists impose the further rationality requirement of regularity : anything that is possible in an appropriate sense gets assigned positive probability. It is meant to capture a form of open-mindedness and responsiveness to evidence.
But then, perhaps unintuitively, someone who assigns probability 0. Probabilistic coherence plays much the same role for degrees of belief that consistency plays for ordinary, all-or-nothing beliefs. It seems, then, that the subjectivist needs something more. And various subjectivists offer more. This resonates with more recent proposals e.
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Since relative frequencies obey the axioms of probability up to finite additivity , it is thought that rational credences, which strive to track them, should do so also. However, rational credences may strive to track various things. For example, we are often guided by the opinions of experts.
We consult our doctors on medical matters, our weather forecasters on meteorological matters, and so on. This idea may be codified as follows:. For example, if you regard the local weather forecaster as an expert on your local weather, and she assigns probability 0. More generally, we might speak of an entire probability function as being such a guide for an agent over a specified set of propositions.
We may go still further. There may be universal expert functions for large classes of rational agents, and perhaps all of them.
Interpretations of Probability (Stanford Encyclopedia of Philosophy)
The Principle of Direct Probability regards the relative frequency function as a universal expert function for all rational agents; we have already seen the importance that proponents of calibration place on it. Hacking :. Lewis posits a similar expert role for the objective chance function, ch , for all rational initial credences in his Principal Principle here simplified [ 8 ] :. For example, a rational agent who somehow knows that a particular coin toss lands heads is surely not required to assign.
The other expert principles surely need to be suitably qualified — otherwise they face analogous counterexamples. Yet strangely, the Principal Principle is the only expert principle about which concerns about inadmissible evidence have been raised in the literature. The ultimate expert, presumably, is the truth function — the function that assigns 1 to all the true propositions and 0 to all the false ones.
So all of the proposed expert probabilities above should really be regarded as defeasible. Joyce portrays the rational agent as estimating truth values, seeking to minimize a measure of distance between them and her probability assignments—that is, to maximize the accuracy of those assignments.
In short, non-probabilistic credences are accuracy-dominated by probabilistic credences. There are some unifying themes in these putative constraints on subjective probability. We have been gradually adding more and more constraints on rational credences, putatively demanded by rationality.
Recall that Carnap first assumed that there was a unique confirmation function, and then relaxed this assumption to allow a plurality of such functions. We now seem to be heading in the opposite direction: starting with the extremely permissive orthodox Bayesianism, we are steadily reducing the class of rationally permissible credence functions. So far the constraints that we have admitted have not been especially evidence -driven. The lines of demarcation are not sharp, and subjective Bayesianism may be regarded as a somewhat indeterminate region on a spectrum of views that morph into objective Bayesianism.
At one end lies an extreme form of subjective Bayesianism, according to which rational credences are constrained only by the probability calculus and updating by conditionalization. But both objective Bayesians and subjective Bayesians may adopt less extreme positions, and typically do. For example, Jon Williamson is an objective Bayesian, but not an extreme one.
He adds to the probability calculus the constraints of being calibrated with evidence, and otherwise equivocating between basic outcomes, especially appealing to versions of maximum entropy. As such, his view is a descendant of the classical interpretation and its generalization due to Jaynes. Gamblers, actuaries and scientists have long understood that relative frequencies bear an intimate relationship to probabilities. Frequency interpretations posit the most intimate relationship of all: identity.
A simple version of frequentism, which we will call finite frequentism , attaches probabilities to events or attributes in a finite reference class in such a straightforward manner:. The crucial difference, however, is that where the classical interpretation counted all the possible outcomes of a given experiment, finite frequentism counts actual outcomes. It is thus congenial to those with empiricist scruples.
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Finite frequentism gives an operational definition of probability, and its problems begin there.