  We study the renormalization group effects on neutrino masses and mixing in
Minimal Supersymmetric Standard Model (MSSM) by considering a $\mu-\tau$
symmetric mass matrix at high energy scale giving rise to Tri-Bi-Maximal (TBM)
type mixing. We outline a flavor symmetry model based on $A_4$ symmetry giving
rise to the desired neutrino mass matrix at high energy scale. We take the
three neutrino mass eigenvalues at high energy scale as input parameters and
compute the neutrino parameters at low energy by taking into account of
renormalization group effects. We observe that the correct output values of
neutrino parameters at low energy are obtained only when the input mass
eigenvalues are large $|m_{1,2,3}| = 0.08-0.12$ eV with a very mild hierarchy
of either inverted or normal type. A large inverted or normal hierarchical
pattern of neutrino masses is disfavored within our framework. We also find a
preference towards higher values of $\tan{\beta}$, the ratio of vacuum
expectation values (vev) of two Higgs doublets in MSSM in order to arrive at
the correct low energy output. Such a model predicting large neutrino mass
eigenvalues with very mild hierarchy and large $\tan{\beta}$ could have
tantalizing signatures at oscillation, neutrino-less double beta decay as well
as collider experiments.
