Bayesianism II: Applications and Criticisms Kenny Easwaran* University of Southern California Abstract In the first paper, I discussed the basic claims of Bayesianism (that degrees of belief are important, that they obey the axioms of probability theory, and that they are rationally updated by either standard or Jeffrey conditionalization) and the arguments that are often used to support them. In this paper, I will discuss some applications these ideas have had in confirmation theory, epistemol- ogy, and statistics, and criticisms of these applications. 1. Bayesian Confirmation Theory One of the important early uses of probability theory in philosophy was in (Carnap 1950), which argued that one notion of probability (corresponding to something like rational degree of belief, though he had in mind a more purely logical notion that gives rise to constraints on degree of belief, rather than working with degree of belief directly (Carnap 1945)) gives the best analysis of the scientific notion of confirmation. Like Hempel before him (Hempel 1945), Carnap sought not just a qualitative theory of whether a given piece E of evidence confirms, disconfirms, or is neutral with respect to a hypothesis H, but also a theory of when E 1 confirms H 1 more strongly than E 2 confirms H 2 , and possibly even an absolute quantitative scale of these degrees of confir- mation. However, Carnap’s book had an unfortunate ambiguity between two different measures of confirmation, pointed out in (Popper 1954). Because this ambiguity was so pervasive in Carnap’s book, he never fixed it, although he acknowledged the problem in the second edition. As it is often put, the ambiguity is between an absolute notion of confirmation as ‘firmness’ (P(H | E)) and an incremental notion of ‘increase in firmness’ (P(H | E) ) P(H)), either one of which can be seen as something like the amount of support E gives to H. Since this has been clarified, most Bayesian measures of confir- mation have aimed at something more like the latter, ‘incremental’ notion, than the former, though the confusion is an easy one to make, and has often been made by others since. Thus, Bayesian confirmation theorists traditionally analyze the qualitative notion of confirmation by saying that E confirms H iff P(H | E)> P(H), though there is still controversy about whether the probability function is the degrees of belief of some particular agent at a time or some other probability function, and whether extra proposi- tions beyond E should be conditionalized on. Bayes’ theorem 1 states that P ðH j EÞ¼ P ðH ÞP ðE j H Þ=P ðEÞ, so that (as usual, assuming for now that none of the relevant probabilities are 0) P(H | E)> P(H) iff P(E | H)> P(E). Given the standard Bayesian picture of update by conditionalization, we can see that if P(H) is the degree of belief a scientist had in H before conducting her experiment, and E is the unique propo- sition she learns as the result of performing the experiment, then P(H | E) will be her degree of belief in H after performing the experiment. Thus, an experiment confirms a Philosophy Compass 6/5 (2011): 321–332, 10.1111/j.1747-9991.2011.00398.x ª 2011 The Author Philosophy Compass ª 2011 Blackwell Publishing Ltd