  Rare decay of the Higgs boson is a powerful tool to probe new physics beyond
the standard model. The rare decays occur through quantum loops in which the
standard model and new physics contributions might cancel each other out. We
consider a faked-no-new-physics scenario that the new physics contributions are
about minus two times the standard model contribution such that partial widths
of Higgs boson rare decays are the same as the standard model predictions. We
propose to measure the $HZ\gamma$ and $H\gamma\gamma$ anomalous couplings in
the process of $e^+ e^- \to H\gamma$ that is strongly correlated to Higgs boson
rare decays. We show that the faked-no-new-physics scenario can be fully probed
at a high energy electron-positron collider.
