Online research in philosophy

According to the Bayesian view, scientific hypotheses must be appraised in terms of their posterior probabilities relative to the available experimental data. Such posterior probabilities are derived from the prior probabilities of the hypotheses by applying Bayes'theorem. One of the most important problems arising within the Bayesian approach to scientific methodology is the choice of prior probabilities. Here this problem is considered in detail w.r.t. two applications of the Bayesian approach: (1) the theory of inductive probabilities (TIP) developed by Rudolf Carnap and other epistomologists and (2) the analysis of the multinational inferences provided by Bayesian statstics (BS). ... Zie: Summary.

Bayesian models of human learning are becoming increasingly popular in cognitive science. We argue that their purported confirmation largely relies on a methodology that depends on premises that are inconsistent with the claim that people are Bayesian about learning and inference. Bayesian models in cognitive science derive their appeal from their normative claim that the modeled inference is in some sense rational. Standard accounts of the rationality of Bayesian inference imply predictions that an agent selects the option that maximizes the posterior expected utility. Experimental confirmation of the models, however, has been claimed because of groups of agents that probability match the posterior. Probability matching only constitutes support for the Bayesian claim if additional unobvious and untested (but testable) assumptions are invoked. The alternative strategy of weakening the underlying notion of rationality no longer distinguishes the Bayesian model uniquely. A new account of rationality—either for inference or for decision-making—is required to successfully confirm Bayesian models in cognitive science.

Because physicians use scientific inference for the generalizations of individual observations and the application of general knowledge to particular situations, the Bayesian probability solution to the problem of induction has been proposed and frequently utilized. Several problems with the Bayesian approach are introduced and discussed. These include: subjectivity, the favoring of a weak hypothesis, the problem of the false hypothesis, the old evidence/new theory problem and the observation that physicians are not currently Bayesians. To the complaint that the prior probability is subjective, Bayesians reply that there will be ultimate convergence, but the rebuttal to this is that there will not be uniform convergence. Secondly, since the Bayesian scheme favors a weak hypothesis, theories turn out to be a gratuitous risk. The problem with the false hypothesis comes out in the denominator of the theorem, revealing that a factor which is not a theory at all is being considered in the reasoning. On the old evidence/new theory problem old evidence cannot confirm a new theory so that the posterior probability will equal the prior probability. Finally, empiric studies have shown that current physicians are not Bayesians. But on consideration of Bayesian inference as a system of inference, it can be reasoned that physicians should be Bayesians. However, the problem of physicians' and patients' own subjectivity continue to plague this system of medical decision making.

Orthodox Bayesian decision theory requires an agent’s beliefs representable by a real-valued function, ideally a probability function. Many theorists have argued this is too restrictive; it can be perfectly reasonable to have indeterminate degrees of belief. So doxastic states are ideally representable by a set of probability functions. One consequence of this is that the expected value of a gamble will be imprecise. This paper looks at the attempts to extend Bayesian decision theory to deal with such cases, and concludes that all proposals advanced thus far have been incoherent. A more modest, but coherent, alternative is proposed. Keywords: Imprecise probabilities, Arrow’s theorem.

Bayesianism is a collection of positions in several related fields, centered on the interpretation of probability as something like degree of belief, as contrasted with relative frequency, or objective chance. However, Bayesianism is far from a unified movement. Bayesians are divided about the nature of the probability functions they discuss; about the normative force of this probability function for ordinary and scientific reasoning and decision making; and about what relation (if any) holds between Bayesian and non-Bayesian concepts.

This essay answers the “Bayesian Challenge,” which is an argument offered by Bayesians that concludes that belief is not relevant to rational action. Patrick Maher and Mark Kaplan argued that this is so because there is no satisfactory way of making sense of how it would matter. The two ways considered so far, acting as if a belief is true and acting as if a belief has a probability over a threshold, do not work. Contrary to Maher and Kaplan, Keith Frankish argued that there is a way to make sense of how belief matters by introducing a dual process theory of mind in which decisions are made at the conscious level using premising policies . I argue that Bayesian decision theory alone shows that it is sometimes rational to base decisions on beliefs; we do not need a dual process theory of mind to solve the Bayesian Challenge. This point is made clearer when we consider decision levels : acting as if a belief is true is sometimes rational at higher decision levels.

Bayesian epistemology addresses epistemological problems with the help of the mathematical theory of probability. It turns out that the probability calculus is especially suited to represent degrees of belief (credences) and to deal with questions of belief change, confirmation, evidence, justification, and coherence. Compared to the informal discussions in traditional epistemology, Bayesian epis- temology allows for a more precise and fine-grained analysis which takes the gradual aspects of these central epistemological notions into account. Bayesian epistemology therefore complements traditional epistemology; it does not re- place it or aim at replacing it.

Marc Lange has criticized my assertion that relative to a Bayesian conception of inductive reasoning, Hume's argument for inductive scepticism cannot be run. I reply that the way in which Lange suggests one should run the Humean argument in a Bayesian framework ignores the fact that in Bayesian models of learning from experience, the domain of an agent's probability measure is exogenously determined. I also show that Lange is incorrect to equate probability distributions which 'support inductive inferences' with probability distributions which assign probability to contingent propositions/events.

The objective theory of probability of Richard von Mises has been criticized by Crovelli (2009), who defends a subjective approach. This paper attempts to clarify the different meanings of ‘objective’ and ‘subjective’ when applied to probability, and then argues for an objective Bayesian theory of probability, as exemplified in the writings [...].

I present a formalism that combines two methodologies: objective Bayesianism and Bayesian nets. According to objective Bayesianism, an agent’s degrees of belief (i) ought to satisfy the axioms of probability, (ii) ought to satisfy constraints imposed by background knowledge, and (iii) should otherwise be as non-committal as possible (i.e. have maximum entropy). Bayesian nets offer an efficient way of representing and updating probability functions. An objective Bayesian net is a Bayesian net representation of the maximum entropy probability function.