One umbrella idea, aimed at different targets each time it's used
Partially specified models start from one worked linear-regression example, and the paper's own citations show how far a single idea can be stretched once other researchers pick it up. Trading the scope of a guarantee for the ability to check it ends up answering several different questions, not one.
Covariate shift's untestable assumption opens the stretch, followed by the generalized method of moments aiming at parameter identification. Unsupervised risk estimation closes it by asking an entirely different question: not what the right model is, but how wrong this one already is.
Partially specified models are introduced through one worked linear-regression example, and the paper's own citations show how much the umbrella stretches. Covariate shift's untestability motivates a retreat to partially specified models generally, trading the scope of a guarantee for the ability to check whether its assumption held. The generalized method of moments generalizes that founding example toward parameter identification, answering "what is the true model." Unsupervised risk estimation partially specifies error rather than parameters, answering the different question "how wrong is this model" without claiming to know the right one, which is also why the generalized method of moments diverges in target from unsupervised risk estimation despite both being partial-specification techniques cited in the same passage. Covariate shift's single invariant assumption also contrasts directly with training on multiple distributions, which asserts nothing checkable at all. The pairing shows partial specification is a strategy, not a single method.