While the tumorigenic effects of specific recurrent mutations in known cancer driver-genes is well-characterized, not much is known about the functional relevance of the vast majority of recurrent mutations observed across cancers. Prior studies have attempted to identify functional genomic aberrations by integrating multi-omics measurements in cancer samples with community-curated biological pathway networks. However, the majority of these approaches overlook the following biological considerations: i) signaling pathway networks are highly tissue-specific and their regulatory interactions differ across tissue types; ii) regulatory factors exhibit heterogeneous influence on downstream gene transcription; iii) epigenetic and genomic alterations exhibit nonlinear impact on gene transcription. In order to accommodate these biological effects, we propose a hybrid Bayesian method to learn tissue-specific pairwise influence models amongst genes and to predict a gene's expression level as a nonlinear-function of its epigenetic and regulatory influences. We employ a novel tree-based depth-penalization mechanism in order to capture the higher regulatory impact of closer neighbors in the regulatory network. Using a breast cancer multi-omics dataset (N=1190), we show that our proposed method has superior prediction power over optimization-based regression models, with the additional advantage of revealing gene deregulations potentially driven by somatic mutations.