  A highly physical model of the subtracted $I=1$ vector polarization, obtained
using a dispersive representation with precise hadronic $\tau$ decay data as
input, is used to investigate systematic issues in the lattice evaluation of
the leading order hadronic vacuum polarization contribution to the anomalous
magnetic moment of the muon. The model is also employed to study possible
resolutions of these problems. A hybrid approach to analyzing lattice data,
involving low-order Pad\'e, low-degree conformal-variable polynomial, or
supplemented NNLO ChPT fits for $Q^2$ below $\sim 0.1-0.2$ GeV$^2$ and direct
numerical integration of lattice data above this point, is shown to bring the
systematic issues identified under control at the sub-$1\%$ level.
