Scientific method

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Bayesian Methods
Central to the Bayesian methods is epistemic probability, a rational agent’s degree of belief. A number of arguments have been put forward to defend the probability axioms as coherence conditions for rational degrees of belief, in analogy to the way logical consistency can be taken as a coherence condition for rational acceptance. Dutch book arguments have shown that degrees of belief violating the probability axioms would assign positive expectations to each bet in a system of bets and conditional bets that would result in sure loss if they were all made together. A number of other arguments for this synchronic condition on rational degrees of belief have been advanced (Ramsey, Savage, Shimony , van Fraassesn, Cox, Good, Aczel).
David Lewis (1941-2001) provided a diachronic Dutch book argument (reported in Teller 1976) to defend the Bayesian conditionalization learning model, according to which assigning new degrees of belief given by

P’ (B) = P(B&A)/P(A)

is the appropriate response to a learning experience in which the total relevant empirical input is to accept A as new evidence. Bas van Fraassen (1941-) extended this diachronic Dutch book argument to defend a condition he called “reflection” (1984). His proposal to treat the reflection condition as a constraint on degrees of belief that could be counted as rational has led to much controversy.
One central Bayesian theme has been to investigate conditions under which evidence leads to convergence of opinion. Bruno de Finetti (1906-1985) specified conditions (exchangeability, absolute mutual continuity) that with subjective probability 1 [? O.S.] would lead Bayesian agents who update by repeated conditionlization on the outcomes of the same observations to converge toward agreement in their degrees of belief, however otherwise divergent their prior degrees of belief may have been ([1937] 1980). Brian Skyrms has given what is probably the most general possible version of de Finetti’s condition for convergence (1990).
Wayne Myrvold (1963-) has argued that for Bayesians the degree to which a hypothesis unifies phenomena contributes to the degree to which these phenomena support the hypothesis (2003). This suggests that Bayesians can recover important aspects of Newton’s method. It may well be that investigating the representation of Newton’s method of provisional acceptance in a Bayesian model will result in enriching the Bayesian framework to make it offer more resources for illuminating scientific method.

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