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Unformatted text preview: Notes for Week 12 of Confirmation 11/28/07 Branden Fitelson 1 Some Background on Subjective Bayesian Confirmation Theory 1.1 Probabilism: Epistemic vs Psychological Subjective Bayesians assume that epistemic/actual agents φ / ψ have degrees of confidence or degrees of belief in propositions, and that these degrees of belief satisfy the probability axioms . In the case of actual agents ψ , this is surely a strong idealization , since it requires a fair amount of logical omniscience (as well as other qualities that actual agents don’t, in general, possess). In the case of epistemic agents, probabilism is also a substantive claim, which is non-trivial to justify. In the “conjunction fallacy” (CF), both normative and descriptive questions are raised, and it is difficult to cleanly separate the two sorts of questions. I will suggest that confirmation may have a role in explaining (if not justifying) CF-responses of actual ψ ’s. 1.2 Background on Bayesian Relevance Measures of Confirmation 1.2.1 The Plethora of Bayesian Relevance Measures As I have mentioned briefly before, there are various quantitative measures c of (relevance) confirmation. Many such measures have been proposed and/or defended in the Bayesian literature (my dissertation Studies in Bayesian Confirmation Theory is all about this plethora of measures). All such measures (when properly understood) will have several things in common. First, they are all relevance measures. That is, they are all sensitive to probabilistic relevance. There are various ways to make this precise. One way is as follows: ( R ) c (H,E | K) > 0 if Pr (H | E & K) > Pr (H | K), < 0 if Pr (H | E & K) < Pr (H | K), = 0 if Pr (H | E & K) = Pr (H | K). This characterization of the relevance requirement presupposes that the measures are properly scaled , so as to respect the sign conventions in ( R ). To simplify matters, I will present all the relevance measures I will be discussing on a normalized [- 1 , 1 ] scale . This can always be done, since all we care about (here) is the comparative (ordinal) structure of relevance measures. The precise numbers they assign will not be important for us. Before presenting and discussing some relevance measures, a definition is in order: Definition . Two measures c 1 (H,E | K) and c 2 (H,E | K) of the degree to which E confirms H relative to K are said to be ordinally equivalent ( c 1 ≈ c 2 ) just in case, for all H , E , K , H , E , K : c 1 (H,E | K) ≥ c 1 (H ,E | K ) iff c 2 (H,E | K) ≥ c 2 (H ,E | K ). As it turns out, all of the measures c in the historical literature can be converted into ordinally equivalent measures c , which are on a normalized [- 1 , 1 ] scale . I will give some examples below, and I will always as- sume we are working with normalized relevance measures, in this sense. The plethora is generated because there are many logically equivalent ways of saying “ E and H are positively correlated, given K ”. For instance:”....
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This note was uploaded on 08/01/2008 for the course PHIL 290 taught by Professor Fitelson during the Fall '06 term at Berkeley.
- Fall '06