  We analyze the stellar initial mass functions (IMF) of a large sample of
early type galaxies (ETGs) provided by MaNGA. The large number of IFU spectra
of individual galaxies provide high signal-to-noise composite spectra that are
essential for constraining IMF and to investigate possible radial gradients of
the IMF within individual galaxies. The large sample of ETGs also make it
possible to study how the IMF shape depends on various properties of galaxies.
We adopt a novel approach to study IMF variations in ETGs, use Bayesian
inferences based on full spectrum fitting. The Bayesian method provides a
statistically rigorous way to explore potential degeneracy in spectrum fitting,
and to distinguish different IMF models with Bayesian evidence. We find that
the IMF slope depends systematically on galaxy velocity dispersion, in that
galaxies of higher velocity dispersion prefer a more bottom-heavy IMF, but the
dependence is almost entirely due to the change of metallicity, $Z$, with
velocity dispersion. The IMF shape also depends on stellar age, $A$, but the
dependence is completely degenerate with that on metallicity through a
combination $AZ^{-1.42}$. Using independent age and metallicity estimates we
find that the IMF variation is produced by metallicity instead of age. The IMF
near the centers of massive ETGs appears more bottom-heavy than that in the
outer parts, while a weak opposite trend is seen for low-mass ETGs.
Uncertainties produced by star formation history, dust extinction,
$\alpha$-element abundance enhancement and noise in the spectra are tested.
