  We study the uncertainties of the Higgs boson production cross section
through the gluon fusion subprocess at the LHC (with $\sqrt s=7, 8$ and $14$
TeV) arising from the uncertainties of the parton distribution functions (PDFs)
and of the value of the strong coupling constant $\alpha_s(M_Z)$. These
uncertainties are computed by two complementary approaches, based on the
Hessian and the Lagrange Multiplier methods within the CTEQ-TEA global analysis
framework. We find that their predictions for the Higgs boson cross section are
in good agreement. Furthermore, the result of the Lagrange Multiplier method
supports the prescriptions we have previously provided for using the Hessian
method to calculate the combined PDF and $\alpha_s$ uncertainties, and to
estimate the uncertainties at the $68\%$ confidence level by scaling them from
the $90\%$ confidence level.
