Likelihoodists generally admit that likelihood functions are insufficient for addressing questions about what to believe or do. Because this admission seems to be an obvious difficulty for their view, one might expect to find arguments in their writings for nevertheless regarding likelihoodism as an adequate self-contained methodology for science. But the arguments of that kind that they provide are weak and few in number.
Although he laid much of the conceptual groundwork for likelihoodism, Hacking was not himself a likelihoodist1 even when he was writing his 1965 Logic of Statistical Inference. In that book he coins the term “Law of Likelihood” for what we would now call the qualitative component of the Law of Likelihood. However, he explicitly denies the usefulness of that principle for applications, writing that it “has received little attention from statisticians because it is hardly ever of practical importance” (63).
Unlike Hacking, Edwards is a likelihoodist. However, he too fails to argue for regarding likelihoodism as an adequate self-contained methodology for science. Edwards seems to fail to see the need for such an argument because he fails to make a clear distinction between a theory of evidential support and a theory of belief or inference. His failure to make such a distinction can be seen in the fact that he states the Law of Likelihood in terms of support (1972, 30) but describes it as providing the basis for a system of inference (e.g. 7) and relative degrees of belief (e.g. 28) without discussion or argument.
Unlike Edwards, Royall is admirably clear and consistent in distinguishing among theories of evidence, theories of inference, and theories of decision (e.g. 1997, 3). But he too fails to argue for regarding likelihoodism as an adequate self-contained methodology for science. Royall seems simply to take for granted that a defensible theory of evidence would be valuable in its own right. He starts his book in support of likelihoodism (1997) with the assertion that the “most important task” of statistics “is to provide objective quantitative alternatives to personal judgement for interpreting the evidence produced by experiments and observational studies” (xi). That assertion has significant prima facie plausibility. However, it could derive that plausibility from the implicit assumption that a theory that allowed for objective interpretations of data as evidence would provide useful guidance for belief or action. And perhaps likelihoodists methods do provide such guidance, but by likelihoodists’ own admissions they do so only when embedded within some other approach such as a subjective Bayesian methodology. Thus, the prima facie plausibility of the claim that providing methods for interpreting data as evidence is an important task of statistics fails to vindicate likelihoodism as an adequate self-contained methodology for science.
Unlike Royall, Sober recognizes that it is not enough to show that the Law of Likelihood “conforms to, and renders precise and systematic, our use of the informal concept” of evidential favoring (2008, 35). As he puts it,
Something important would be missed if the Law of Likelihood were judged solely on the basis of how it clarifies the meaning of the English word “likely”…. What matters about the Law of Likelihood is whether it isolates an epistemologically important concept…. A formal proposal that describes how an informal concept should be understood is to be judged by the light it throws on the informal concept, but it also should be judged by the light it throws, period. (2008, 35)
Sober’s concern seems to be less specific than mine in that a broad range of payoffs for the Law of Likelihood would satisfy it (such as certain applications in cognitive science2), whereas I am specifically looking for a methodological payoff. But he at least recognizes something like the difficulty for likelihoodism that concerns me here. However, Sober gives no explicit response to that difficulty; after the passage just quoted he immediately moves on to discussing purported counterexamples to the Law of Likelihood, and he never returns to the issue. He clearly takes himself to vindicate likelihoodism for some applications in the course of his book, so he must take himself to show that the Law of Likelihood does in fact isolate an epistemologically important concept somehow.
Perhaps Sober’s implicit response to the difficulty under discussion is in his use of the Law of Likelihood to address seemingly well-formed and important questions such as the following (Sober 2008, 107-8):
- Are the imperfect adaptations that organisms exhibit evidence that they were not produced by an intelligent designer?
- Is the fact that bears in cold climates have longer fur than bears in warm climates evidence that fur length evolved by natural selection as an adaptive response to ambient temperature?
- Are the similarities that species exhibit evidence that they stem from a common ancestor?
The fact that the Law of Likelihood answers such questions might seem sufficient to establish its usefulness, assuming that it answers them correctly. However, it does so only if answers to questions about evidential favoring are valuable in themselves, apart from any connection they might have to questions about belief or action.
Likelihoodists have provided no account of what we are to do with answers to questions about evidential favoring if not to use them to address questions about belief or action and have provided no purely likelihood-based account of how we are to use them to answer such questions. One could maintain that we are simply to contemplate, report, and discuss such answers, but it is hard to see what use science would be to society if that were are all scientists could legitimately do.
- Am I being fair to likelihoodism?
- What is your best defense of the view that it is possible to provides an adequate self-contained methodology for science on the basis of likelihood functions alone?
To share your thoughts about this post, comment below or send me an email. To use $\LaTeX$ in comments, surround mathematical expressions with single dollar signs for inline mode or double dollar signs for display mode.