  We investigate Schmidt's conjecture (i.e., that the star formation rate
scales in a power-law fashion with the gas density) for four well-studied local
molecular clouds (GMCs). Using the Bayesian methodology we show that a local
Schmidt scaling relation of the form Sigma*(A_K) = kappa x (A_K)^{beta}
(protostars pc^{-2}) exists within (but not between) GMCs. Further we find that
the Schmidt scaling law, by itself, does not provide an adequate description of
star formation activity in GMCs. Because the total number of protostars
produced by a cloud is given by the product of Sigma*(A_K) and S'(> A_K), the
differential surface area distribution function, integrated over the entire
cloud, the cloud's structure plays a fundamental role in setting the level of
its star formation activity. For clouds with similar functional forms of
Sigma*(A_K), observed differences in their total SFRs are primarily due to the
differences in S'(> A_K) between the clouds. The coupling of Sigma*(A_K) with
the measured S'(> A_K) in these clouds also produces a steep jump in the SFR
and protostellar production above A_K ~ 0.8 magnitudes. Finally, we show that
there is no global Schmidt law that relates the star formation rate and gas
mass surface densities between GMCs. Consequently, the observed
Kennicutt-Schmidt scaling relation for disk galaxies is likely an artifact of
unresolved measurements of GMCs and not a result of any underlying physical law
of star formation characterizing the molecular gas.
