  We establish a general Bernstein--von Mises theorem for approximately linear
semiparametric functionals of fractional posterior distributions based on
nonparametric priors. This is illustrated in a number of nonparametric settings
and for different classes of prior distributions, including Gaussian process
priors. We show that fractional posterior credible sets can provide reliable
semiparametric uncertainty quantification, but have inflated size. To remedy
this, we further propose a \textit{shifted-and-rescaled} fractional posterior
set that is an efficient confidence set having optimal size under regularity
conditions. As part of our proofs, we also refine existing contraction rate
results for fractional posteriors by sharpening the dependence of the rate on
the fractional exponent.
