  The recent Cosmic Microwave Background (CMB) measurements indicate that there
is power deficiency of the CMB anisotropies at large scales compared with the
$\Lambda$CDM model. We have investigated the possibility of explaining such
effects by a class of primordial power spectra which have infrared cutoffs
close to the horizon scale. The primordial power spectrum recovered by direct
deconvolution of the observed CMB angular spectrum indicates that the data
prefers a sharp infrared cutoff with a localized excess (bump) just above the
cutoff. We have been motivated to assess plausible extensions of simplest
inflationary scenarios which readily accommodate similar form of infrared
cutoff. We carry out a complete Bayesian analysis of the parameter space using
{\it Markov Chain Monte Carlo} technique with such a class of primordial power
spectra. We show that primordial power spectrum that have features such as an
infrared cutoff followed by a subsequent excess in power give better fit to the
observed data compared to a nearly scale-invariant power law or power spectrum
with just a monotonic infrared cutoff. However, there is substantial room for
improvement in the match to data and calls for exploration of other mechanisms
that may lead to infrared cutoff even closer to that recovered by direct
deconvolution approach.
