Sparsity is a standard structural assumption that is made while modeling high-dimensional
statistical parameters. This assumption essentially entails a lower dimensional embedding of the
high-dimensional parameter thus enabling sound statistical inference. Apart from this obvious
statistical motivation, in many modern applications of statistics such as Genomics, Neuroscience
etc. parameters of interest are indeed of this nature.
For over almost two decades, spike and slab type priors have been the Bayesian gold standard
for modeling of sparsity. However, due to their computational bottlenecks shrinkage priors have
emerged as a powerful alternative. This family of priors can almost exclusively be represented
as a scale mixture of Gaussian distribution and posterior Markov chain Monte Carlo (MCMC)
updates of related parameters are then relatively easy to design. Although shrinkage priors were
tipped as having computational scalability in high-dimensions, when the number of parameters is
in thousands or more, they do come with their own computational challenges. Standard MCMC
algorithms implementing shrinkage priors generally scale cubic in the dimension of the parameter
making real life application of these priors severely limited. The first chapter of this document
addresses this computational issue and proposes an alternative exact posterior sampling algorithm
complexity of which that linearly in the ambient dimension.
The algorithm developed in the first chapter is specifically designed for regression problems.
However, simple modifications of it allows tackling other high-dimensional problems where these
priors have found little application. In the second chapter, we develop a Bayesian method based
on shrinkage priors for high-dimensional multiple response response regression. We show how
proper shrinkage may be used for modeling high-dimensional low-rank matrices. Unlike spike
and slab type priors, shrinkage priors are unable to produce exact zeros in the posterior. In this
chapter we also devise two independent post MCMC processing schemes based on the idea of
soft-thresholding with default choices of tuning parameters. This post processing steps provide
exact estimates of the row and rank sparsity in the parameter matrix.
Theoretical study of the posterior convergence rates using shrinkage priors are relatively underdeveloped.
While we do not attempt to provide a unifying foundation to study these properties,
in chapter three we choose a specific member of the shrinkage family known as the horseshoe prior
and study its convergence rates in several high-dimensional models. These results are completely
new in the literature and also establish the horseshoe priors' optimality in the minimax sense in
- Mallick, Bani Distinguished Professor