Resurrecting logical probability (original) (raw)
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Artificial Intelligence, 2004
In the current discussion about the capacity of Bayesianism in reasoning under uncertainty, there is a conceptual and notational confusion between the explicit condition and the implicit condition of a probability evaluation. Consequently, the limitation of Bayesianism is often seriously underestimated. To represent the uncertainty of a belief system where revision is needed, it is not enough to assign a probability value to each belief.
Bayesianism: Objections and Rebuttals
Philosophical Foundations of Evidence Law, 2021
While the laws of probability are rarely disputed, the question of how we should interpret probability judgments is less straightforward. Broadly, there are two ways to conceive of probability—either as an objective feature of the world, or as a subjective measure of our uncertainty. Both notions have their place in science, but it is the latter subjective notion (the Bayesian approach) that is crucial in legal reasoning. This chapter explains the advantages of using Bayesian networks in adjudicative factfinding. It addresses a number of common objections to the Bayesian approach, such as “There is no such thing as a probability of a single specified event”; “The Bayesian approach only works with statistical evidence”; “The Bayesian approach is too difficult for legal factfinders to comprehend”; and “A Bayesian network can never capture the full complexity of a legal case.” Fenton and Lagnado offer rebuttals to each of these objections.
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Philosophy Compass, 2011
Bayesianism is a popular position (or perhaps, positions) in the philosophy of science, epistemology, statistics, and other related areas, which represents belief as coming in degrees, measured by a probability function. In this article, I give an overview of the unifying features of the different positions called 'Bayesianism', and discuss several of the arguments traditionally used to support them.
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How strongly should you believe the various propositions that you can express? That is the key question facing Bayesian epistemology. Subjective Bayesians hold that it is largely (though not entirely) up to the agent as to which degrees of belief to adopt. Objective Bayesians, on the other hand, maintain that appropriate degrees of belief are largely (though not entirely) determined by the agent's evidence. This book states and defends a version of objective Bayesian epistemology. According to this version, objective Bayesianism is characterized by three norms: · Probability - degrees of belief should be probabilities · Calibration - they should be calibrated with evidence · Equivocation - they should otherwise equivocate between basic outcomes Objective Bayesianism has been challenged on a number of different fronts. For example, some claim it is poorly motivated, or fails to handle qualitative evidence, or yields counter-intuitive degrees of belief after updating, or suffers from a failure to learn from experience. It has also been accused of being computationally intractable, susceptible to paradox, language dependent, and of not being objective enough. Especially suitable for graduates or researchers in philosophy of science, foundations of statistics and artificial intelligence, the book argues that these criticisms can be met and that objective Bayesianism is a promising theory with an exciting agenda for further research.
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How to assign numerical values for probabilities that do not seem artificial or arbitrary is a central question in Bayesian statistics. The case of assigning a probability to the truth of a proposition or event for which there is no evidence other than that the event is contingent, is contrasted with the assignment of probability in the case where there is definte evidence that the event can happen in a finite set of ways. The truth of a proposition of this kind is frequently assigned a probability via arguments of ignorance, symmetry, randomness, the Principle of Indiffernce, the Principal Principal, non-informativeness, or by other methods. These concepts are all shown to be flawed or to be misleading. The statistical syllogism introduced by Williams in 1947 is shown to fix the problems that the other arguments have. An example in the context of model selection is given.
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In this paper, we analyse the various meanings of probability and its different applications, and we focus especially on the classical, the frequentist, and the subjectivist view. We describe the different problems of how probability can be measured in each of the approaches, and how each of them can be well justified by a mathematical theory. We analyse the foundations of probability, where the scientific analysis of the theory that allows for a frequentist interpretation leads to unsolvable problems. Kolmogorov’s axiomatic theory does not suffice to establish statistical inference without further definitions and principles. Finally, we show how statistical inference essentially determines the meaning of probability and a shift emerges from purely objectivist views to a complementary conception of probability with frequentist and subjectivist constituents. For didactical purpose, the result of the present analyses explains basic problems of teaching, originating from a biased focus...
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In the first paper, I discussed the basic claims of Bayesianism (that degrees of belief are important, that they obey the axioms of probability theory, and that they are rationally updated by either standard or Jeffrey conditionalization) and the arguments that are often used to support them. In this paper, I will discuss some applications these ideas have had in confirmation theory, epistemology, and statistics, and criticisms of these applications.
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