  As Basu (1977) writes, "Eliminating nuisance parameters from a model is
universally recognized as a major problem of statistics," but after more than
50 years since Basu wrote these words, the two mainstream schools of thought in
statistics have yet to solve the problem. Fortunately, the two mainstream
frameworks aren't the only options. This series of papers rigorously develops a
new and very general inferential model (IM) framework for
imprecise-probabilistic statistical inference that is provably valid and
efficient, while simultaneously accommodating incomplete or partial prior
information about the relevant unknowns when it's available. The present paper,
Part III in the series, tackles the marginal inference problem. Part II showed
that, for parametric models, the likelihood function naturally plays a central
role and, here, when nuisance parameters are present, the same principles
suggest that the profile likelihood is the key player. When the likelihood
factors nicely, so that the interest and nuisance parameters are perfectly
separated, the valid and efficient profile-based marginal IM solution is
immediate. But even when the likelihood doesn't factor nicely, the same
profile-based solution remains valid and leads to efficiency gains. This is
demonstrated in several examples, including the famous Behrens--Fisher and
gamma mean problems, where I claim the proposed IM solution is the best
solution available. Remarkably, the same profiling-based construction offers
validity guarantees in the prediction and non-parametric inference problems.
Finally, I show how a broader view of this new IM construction can handle
non-parametric inference on risk minimizers and makes a connection between
non-parametric IMs and conformal prediction.
