  We re-examine the predictiveness of single-field inflationary models and
discuss how an unknown UV completion can complicate determining inflationary
model parameters from observations, even from precision measurements. Besides
the usual naturalness issues associated with having a shallow inflationary
potential, we describe another issue for inflation, namely, unknown UV physics
modifies the running of Standard Model (SM) parameters and thereby introduces
uncertainty into the potential inflationary predictions. We illustrate this
point using the minimal Higgs Inflationary scenario, which is arguably the most
predictive single-field model on the market, because its predictions for $A_s$,
$r$ and $n_s$ are made using only one new free parameter beyond those measured
in particle physics experiments, and run up to the inflationary regime. We find
that this issue can already have observable effects. At the same time, this
UV-parameter dependence in the Renormalization Group allows Higgs Inflation to
occur (in principle) for a slightly larger range of Higgs masses. We comment on
the origin of the various UV scales that arise at large field values for the SM
Higgs, clarifying cut off scale arguments by further developing the formalism
of a non-linear realization of $\rm SU_L(2) \times U(1)$ in curved space. We
discuss the interesting fact that, outside of Higgs Inflation, the effect of a
non-minimal coupling to gravity, even in the SM, results in a non-linear EFT
for the Higgs sector. Finally, we briefly comment on post BICEP2 attempts to
modify the Higgs Inflation scenario.
