I am considering reframing my paper-in-progress “Why I Am Not a Likelihoodist” around the following argument:
- The output of a likelihoodist method is a characterization of data as evidence.
- The only good use for a characterization of data as evidence is as an input for a justified inference or decision procedure.
- An inference or decision procedure is justified only if it at least approximates a reasonable frequentist or Bayesian procedure.
- Therefore, the only good use for the output of a likelihoodist method is as an input for an inference or decision procedure that at least approximates a reasonable Bayesian or frequentist procedure.
Premise 3 seems to be the most contentious. Here is a first-pass argument for it.1 Suppose we are faced with the task of choosing a rule $\delta(X)$ for making an inference about some parameter $\theta$ based on the value of some random variable $X$, where the relevant loss function for the resulting inference is $l(\delta(X),\theta)$. The choice among different possible decision rules $\delta(X)$ should depend only on the expectation of $l(\delta(X),\theta)$ as a functional of $\delta(X)$. There are two ways to get such an expectation: ascribe a probability distribution to $X$ as a function of $\theta$ and fix $\theta$, or ascribe a probability distribution to $\theta$ and condition on the observed value of $X$. The first way leads to frequentism, the second to Bayesianism. Thus, frequentism and Bayesianism exhaust the reasonable options.
This argument is compatible with the view that it is useful to report the output of a likelihoodist method so that individual recipients of one’s report can use it to update their subjective degrees of belief. However, the process of updating one’s subjective degrees of belief in this way is not straightforward and can lead to misleading results if done in a naive way (see this post).
Discussion Question: What do you think about this argument?
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