  Polygenic risk scores (PRS) developed from genome-wide association studies
(GWAS) are of increasing interest for clinical and research applications.
Bayesian methods have been popular for building PRS because of their natural
ability to regularize models and incorporate external information. In this
article, we present new theoretical results, methods, and extensive numerical
studies to advance Bayesian methods for PRS applications. We identify a
potential risk, under a common Bayesian PRS framework, of posterior impropriety
when integrating the required GWAS summary-statistics and linkage
disequilibrium (LD) data from two distinct sources. As a principled remedy to
this problem, we propose a projection of the summary statistics data that
ensures compatibility between the two sources and in turn a proper behavior of
the posterior. We further introduce a new PRS method, with accompanying
software package, under the less-explored Bayesian bridge prior to more
flexibly model varying sparsity levels in effect size distributions. We
extensively benchmark it against alternative Bayesian methods using both
synthetic and real datasets, quantifying the impact of both prior specification
and LD estimation strategy. Our proposed PRS-Bridge, equipped with the
projection technique and flexible prior, demonstrates the most consistent and
generally superior performance across a variety of scenarios.
