  We propose a model of brain atrophy as a function of high-dimensional genetic
information and low dimensional covariates such as gender, age, APOE gene, and
disease status. A nonparametric single-index Bayesian model of high dimension
is proposed to model the relationship with B-spline series prior on the unknown
functions and Dirichlet process scale mixture of centered normal prior on the
distributions of the random effects. The posterior rate of contraction without
the random effect is established for a fixed number of regions and time points
with increasing sample size. We implement an efficient computation algorithm
through a Hamiltonian Monte Carlo (HMC) algorithm. The performance of the
proposed Bayesian method is compared with the corresponding least square
estimator in the linear model with horseshoe prior, LASSO and SCAD penalization
on the high-dimensional covariates. The proposed Bayesian method is applied to
a dataset on volumes of brain regions recorded over multiple visits of 748
individuals using 620,901 SNPs and 6 other covariates for each individual, to
identify factors associated with brain atrophy.
