Abstract
The foundations of a detailed grammar of Bayesian confirmation are presented as a theoretical tool for the formal analysis of reasoning in epistemology and philosophy of science. After a discussion of core intuitions grounding the measurement of confirmation in probabilistic terms, a number of basic, derived and structural properties of Bayesian incremental confirmation are defined, distinguished and investigated in their logical relationships. Illustrations are provided that a thorough development of this line of research would yield an appropriate general framework of inquiry for several analyses and debates surrounding confirmation and Bayesian confirmation in particular.
Research supported by a grant from the Spanish Department of Science and Innovation (FFI2008-01169/FISO).
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- 1.
It should be kept in mind that this relationship is strictly speaking a three-place one, involving a given background of knowledge and assumptions, often denote as K. Such a term will be omitted from our notation for simple reasons of convenience, as it is unconsequential for our discussion.
- 2.
One can define a rather natural odds counterpart of the measure c rd (H,E) of the relative reduction of probability distance (from certainty). (An earlier occurrence of this measure appears in Heckerman (1986).) As shown in Crupi (2008), however, such an odds counterpart turns out to be ordinally equivalent to the simple odds ratio measure c or (H,E).
- 3.
There are confirmation measures whose behavior is perfectly defined and identical for any p-normal pair while being divergent for non-p-normal ones. To illustrate, consider the measure advocated by Kuipers (2000), i.e., \({c}_{k}(\mathit{H,E}) = p(E\vert H)/p(E)\). Since for any p-normal pair \(p(E\vert H)/p(E) = p(H\vert E)/p(H)\), in this class of cases c k (H,E) is identical to the probability ratio measure c r (H,E) defined above. However, the latter is not defined whenever p(H) = 0. On the contrary, c k (H,E) may be defined in this case as well, provided that E is p-normal and a value for p(E | H) can be specified. (For more on this point, see the distinction between “inclusive” and “non-inclusive” accounts of confirmation in Kuipers (2000); also see Festa (1999, pp 67–68).)
- 4.
The Appendix provides proofs of this as well as all subsequent theorems.
- 5.
(PS) essentially amounts to a statement that Steel (2007) labels LP1 and identifies as one among two possible renditions of the “likelihood principle”. While departing from his terminological choices, we concur with Steel’s argument that (PS) is a compelling principle for Bayesians.
- 6.
We borrow the term “surprise bonus” from Kuipers (2000, 55).
- 7.
- 8.
- 9.
(LL) essentially amounts to a statement that Steel (2007) labels LP2 and identifies as the second possible renditions of the “likelihood principle”. (See footnote 5.)
- 10.
A major issue in Crupi et al. (2007, 236–242) is a thorough analysis of so-called “symmetries and asymmetries” in Bayesian confirmation theory (see Eells and Fitelson 2002). In our current terms, their convergent symmetries are all ordinal structural conditions, whereas their divergent ones are all quantitative structural conditions.
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Crupi, V., Festa, R., Buttasi, C. (2009). Towards a Grammar of Bayesian Confirmation. In: Suárez, M., Dorato, M., Rédei, M. (eds) EPSA Epistemology and Methodology of Science. Springer, Dordrecht. https://doi.org/10.1007/978-90-481-3263-8_7
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