frequentist epiphany

I woke up today wondering what would a frequentist do?, especially with regards to a fitting problem with nuisance parameters; that is, a problem where there are parameters (m,b) that one cares about, and parameters (P,Y,V) that one doesn't. A strict frequentist has no measure on the parameter space, so he or she can't marginalize or integrate out the nuisance parameters from the answer. Also, the frequentist does not produce a probability distribution for the parameters given the data, but rather a probability for the data given the parameters (a likelihood, which has the wrong units to be marginalized). Marginalization is not an option.

I think a principled frequentist has two choices: Either choose the maximum-likelihood model, and report all five parameters, even though three are uninteresting, and then do some kind of sampling-based error analysis around that peak; or else look at the entire space of models that are above some threshold in likelihood. If the latter, the frequentist gets out a range of possibilities for all five parameters. That range is a strict range, depending only on the likelihood threshold. The fact that some parameters are nuisance parameters is irrelevant. The frequentist is more ad hoc because a threshold must be chosen, but also more conservative because in general the prior and the marginalization (at least slightly) tighten up the parameter bounds. (Note that I am assuming that the Bayesian sets priors objectively, which is often not the case.)

1 comment:

  1. You should make some "What Would A Frequentist Do?" t-shirts.