Bayesian Philosophy of Science Sprenger Hartmann Gems Sets
Bayesian Philosophy of Science Sprenger & Hartmann
Gems Sets up conditions of adequacy and corresponding representation theorems for measures of confirmation Motivates the Bayesian view well; includes convincing examples that show Bayesian Confirmation Theory can be used by both historians of science and practicing scientists (descriptive and normative), and solves well-known paradoxes of induction Embraces confirmational pluralism and an understanding of the limitations of Bayesian confirmation; compatible with a Material Theory of Induction
Motivating a probabilistic confirmation theory 1. Probability is a guide to life, and confirmation is a guide to probability Ex. Beach trip depends on probability of rain 2. Probability is the preferred tool for expressing uncertainty in science Ex. Gaussian noise distribution 3. Statistics is formulated in terms of probability Ex. de Finetti’s representation theorem
Bayesian Confirmation Theory
Bayesian Confirmation Theory Prior-Posterior Dependence Final Probability Incrementality Qualitative. Quantitative Bridge Principle
Prior-Posterior Dependence
Final Probability Incrementality
Qualitative-Quantitative Bridge Principle
Bayesian Confirmation Theory Prior-Posterior Dependence Final Probability Incrementality Confirmation as firmness of belief Qualitative. Quantitative Bridge Principle Confirmation as increase in firmness of belief
Bayesian Confirmation Theory Prior-Posterior Dependence Final Probability Incrementality Confirmation as firmness of belief Local Equivalence Requirement of Total Evidence Theorem 1. 1: Confirmation as firmness ? Qualitative. Quantitative Bridge Principle
Local Equivalence
Confirmation as firmness
Confirmation as firmness
Paradox of tacking by conjunction
Requirement of Total Evidence
Bayesian Confirmation Theory Prior-Posterior Dependence Final Probability Incrementality Qualitative. Quantitative Bridge Principle Confirmation as firmness of belief Local Equivalence Requirement of Total Evidence Theorem 1. 1: Confirmation as firmness ? Confirmation as increase in firmness of belief Law of Likelihood Log-ratio measure Modularity Log-Likelihood & Kemeny. Oppenheim measures Disjunction of Alternative Hypotheses Difference measure Contraposition/ Commutativity Generalized entailment measure
Confirmation as increase in firmness
Confirmation as increase in firmness Adequacy conditions Prior-Posterior Dependence & Law of Likelihood Prior-Posterior Dependence & Modularity Prior-Posterior Dependence & Disjunction of Alternative Hypotheses Prior-Posterior Dependence & Contraposition/Commutativity Representation Theorem
Confirmation as increase in firmness We are left with a choice between several confirmation measures of confirmation as increase in firmness. Which should we choose? Confirmational monism: There is a definite answer to this question. Despite the plurality of confirmation measures, there is one that outperforms its competitors. Confirmational pluralism: Which measure performs best depends on a specific context or goals of inquiry. The choice between measures may also depend on empirical findings. “The idea that there is ‘one true measure of confirmation’ is therefore problematic […] there are different senses of degree of confirmation that correspond to different explications. ”
Paradox of the ravens
Paradox of the ravens Bayesian Solution: Bayesian confirmation as increase in firmness has shown us that not all instances of universal conditionals raise their probability. We have to throw out Nicod’s Condition. I. J. Good’s Counter example: World 1 World 2 Black ravens 100 1, 000 Non-black ravens 0 1 Other objects 1, 000, 000
Paradox of the ravens
Goodman’s Grue paradox Problem: Two incompatible hypotheses confirmed by the same piece of evidence Goodman’s Solution: Restrict confirmation to generalizable, projectible predicates which have a successful prediction history Bayesian Solution: 1. Abandon the idea that evidence cannot confirm incompatible hypotheses. 2. The prior probability of green is higher than that of grue, so for confirmation as both firmness and as increase in firmness, green is confirmed to a higher degree than grue That being said…
Goodman’s Grue paradox That being said… “Goodman shows a general problem formal reasoning about confirmation and evidence” there is no viable complete theory of inductive support (see also Norton forthcoming). ” Here, the choice of prior probabilities cannot come from Bayesian reasoning itself. They come from scientific reasoning, and BCT can’t tell you which prior degrees of belief are reasonable.
Gems Sets up conditions of adequacy and corresponding representation theorems for measures of confirmation Motivates the Bayesian view well; includes convincing examples that show Bayesian Confirmation Theory can be used by both historians of science and practicing scientists (descriptive and normative) Embraces confirmational pluralism and an understanding of the limitations of Bayesian confirmation; compatible with a Material Theory of Induction
Discussion 1. Validity of the Bayesian resolutions to: • paradox of tacking by conjunction • black raven paradox • grue paradox 2. Has the Bayesian view of confirmation been redeemed? 3. Bayesian Confirmation Theory and a Material Theory of Induction
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