This post is part of a series (introduced here) in which I present objections to possible responses to the claim that likelihoodism is not a viable alternative to Bayesian and frequentist methodologies because it does not address questions about what to believe or do. I am currently considering the response that likelihoodism is a viable alternative methodology because characterizing one’s data as evidence is valuable in itself. I addressed two arguments for this claim here and here.

A likelihood ratio is a ratio of posterior odds to prior odds under Bayesian updating. Likelihoodists agree with frequentists that prior probabilities that merely represent some individual’s degrees of belief are not appropriate for use in science. Unlike frequentists, they maintain that a likelihood ratio “means the same” as a measure of evidential favoring regardless of whether prior probabilities they regard as legitimate are available or not. But what is a measure of evidential favoring in the likelihoodist’s sense, if not a measure of the shift in one’s degrees of belief that the data warrant?

It seems that the likelihoodist claim that characterizing as evidence is valuable in itself requires an answer to this question. However, no likelihoodist to my knowledge has attempted to provide one. Edwards (1972) and Royall (1997) each address it by arguing in effect that no answer is needed. I responded to Royall’s argument in my previous post. Unfortunately, I do not have access to my copy of Edwards’s book at the moment, so instead of responding to his argument I will consider an objection to the claim that a likelihood ratio “means the same” across different contexts that is due to Ian Hacking.

Hacking coined the phrase “Law of Likelihood” in his (1965), but he used it to refer only to the qualitative claim that *E* favors *H*_{1} over *H*_{2} if Pr(*E*;*H*_{1})> Pr(*E*;*H*_{2}). He did not include the quantitative claim that the likelihood ratio Pr(*E*;*H*_{1})/Pr(*E*;*H*_{2}) is a measure of the degree to which *E* favors *H*_{1} over *H*_{2}. Edwards (1972) seems to have been the first to combine the qualitative and quantitative claims into a single principle, which he called the Likelihood Axiom (31). Today it is standard to follow Royall in using Hacking’s phrase “Law of Likelihood” for the conjunction of the qualitative and quantitative claims.

In a review of (Edwards 1972), Hacking expresses doubts about both the quantitative and the qualitative parts of the Law of Likelihood and argues specifically against the assumption of the quantitative part that a likelihood ratio “means the same” in different contexts (1972, 136). [Read more…]