  We consider the inclusive production of a Higgs boson in gluon-fusion and we
study the impact of threshold resummation at next-to-next-to-next-to-leading
logarithmic accuracy (N$^3$LL) on the recently computed fixed-order prediction
at next-to-next-to-next-to-leading order (N$^3$LO). We propose a conservative,
yet robust way of estimating the perturbative uncertainty from missing higher
(fixed- or logarithmic-) orders. We compare our results with two other
different methods of estimating the uncertainty from missing higher orders: the
Cacciari-Houdeau Bayesian approach to theory errors, and the use of algorithms
to accelerate the convergence of the perturbative series. We confirm that the
best convergence happens at $\mu_R=\mu_F=m_H\,/\,2$, and we conclude that a
reliable estimate of the uncertainty from missing higher orders on the Higgs
cross section at 13 TeV is approximately $\pm4$%.
