  We show that a two-band model with ${\bf k}$-dependent superconducting gaps
well describes the transmission and optical conductivity measured for MgB$_2$
thin films. It is also shown that the two-band anisotropic model consistently
describes the specific-heat jump and thermodynamic critical magnetic field
$H_c$. A single-gap anisotropic model is shown to be insufficient to understand
consistently optical and thermodynamic behaviors. In our model, the pairing
symmetry in each band has an anisotropic characteristic which is determined
almost uniquely; the superconducting gap in the $\sigma$-band has anisotropy in
the ab-plane and the gap in the $\pi$-band has a prolate form exhibiting
anisotropy in the c-direction. The anisotropy in the $\sigma$-band produces
rather small effects on the physical properties compared to the anisotropy in
the $\pi$-band.
