  IIt is known that a statistically significant treatment may not be clinically
significant. A quantity that can be used to assess clinical significance is
called the minimum clinically important difference (MCID), and inference on the
MCID is an important and challenging problem. Modeling for the purpose of
inference on the MCID is non-trivial, and concerns about bias from a
misspecified parametric model or inefficiency from a nonparametric model
motivate an alternative approach to balance robustness and efficiency. In
particular, a recently proposed representation of the MCID as the minimizer of
a suitable risk function makes it possible to construct a Gibbs posterior
distribution for the MCID without specifying a model. We establish the
posterior convergence rate and show, numerically, that an appropriately scaled
version of this Gibbs posterior yields interval estimates for the MCID which
are both valid and efficient even for relatively small sample sizes.
