In the past decades, the emergence of high-density genome-wide single nucleotide polymorphism (SNP) arrays has revolutionized the field of genetic analysis, particularly for complex traits. Linear mixed models that can integrate such large-scale genomic data are widely used for genomic prediction \citep{meuwissen2001prediction,vanraden2008efficient} and genome-wide association studies \citep{visscher201710}.Alongside these, Bayesian approaches have gained significance for their ability to accommodate biologically meaningful prior information on the structure of underlying genetics for genomic prediction and provide probabilistic interpretations of genetic associations.

Recent advancements in high-throughput phenotyping platforms and multi-omics methodologies, coupled with the progress in functional mapping, pave the way for the generation of expansive phenomic, transcriptomic, and metabolomic profiles, along with detailed functional annotations of the genomes. The acquisition of these innovative data sets opens new avenues for us, enabling significant acceleration in enhancing the accuracy of genomic predictions and discoveries of causal variants in genome-wide association studies.

In this thesis, three distinct Bayesian models are developed to address three specific scenarios, each aimed at further improving the accuracy of genome-wide prediction and the power of genome-wide association analyses: 1) The analysis of large-scale longitudinal data; 2) The analysis of high-dimensional and highly correlated (molecular) phenotypic data; and 3) The analysis of multiple traits with comprehensive annotation information on genome.

The first part of this research presents the development and application of a Bayesian random regression model with mixture priors, named RR-BayesC, to analyze time-series measurements in genome-enabled analyses. Based on simulated and real rice (Oryza sativa L.) data, we show that RR-BayesC has a promising ability to distinguish quantitative trait loci (QTL) that are invariant to temporal covariates and QTL that interact with time, resulting in high prediction accuracy even when forecasting is required for phenotypes at later periods.

The second part of this research proposes a high-dimensional Bayesian multivariate regression model with mixture priors, named MegaBayesC, to simultaneously analyze genetic variants underlying thousands of traits. Applied to Genomic Prediction, MegaBayesC effectively integrates hyperspectral reflectance data from 620 hyperspectral wavelengths to improve the accuracy of breeding value prediction on grain yield in a wheat dataset. Applied to Genome-Wide Association Studies, we use simulations to show that MegaBayesC can accurately estimate the effect sizes of QTL across various genetic architectures and causes of correlations among traits. Furthermore, application of MegaBayesC to identify genetic associations with flowering time in Arabidopsis thaliana, leveraging expression data from 20,843 genes is also presented.

The third part of this research proposes an annotation-assisted summary statistics-based multi-trait BayesC model, named MT-SBayesC-func. This model is designed to explore the pleiotropic genetic structure underlying two traits, by incorporating additional information from functional annotations. Through simulation studies, MT-SBayesC-func has demonstrated its ability to accurately estimate annotation-specific genetic correlations and polygenic overlaps, which deepens our understanding of the shared genetic mechanisms underlying complex traits.

With the development of high-throughput sequencing, whole-genome analysis, such as genomic prediction and genome-wide association studies (GWAS), plays an important role in animal and plant breeding. Under the infinitesimal model, complex traits are assumed to be affected by many genes with small additive effects, and the relationship between genotypes and phenotypes is linear. In most GWAS and genomic prediction studies, one goal is to estimate the joint effects of all SNP markers. The landmark paper by Meuwissen et al. (2001) introduced the Bayesian linear mixed model for whole-genome prediction, which has been widely used in breeding programs.

However, as the amount and diversity of omics data continue to grow, several challenges arise for the linear mixed model. First, there is a need to extend mixed models to incorporate multiple sequential layers of data as one connected network (e.g., the regulatory cascades). Second, due to increasing concerns about data privacy, there is a need to adopt mixed models for encrypted data, enabling the sharing of confidential data in genome-to-phenome analyses.

New methods were proposed in this thesis that try to solve these two challenges. For the first challenge, we provide a novel framework named mixed model neural network ("NNMM") to extend the mixed model ("MM") to a multilayer neural network ("NN"), thus incorporating sequential layers of data as a unified multilayer network. Nonlinear relationships between different layers of data are allowed via nonlinear activation functions in neural networks. Moreover, NNMM allows various missing patterns for the data in the middle layer, and the network architecture of NNMM can be predefined to be partially connected.

For the second challenge, a homomorphic encryption method based on high-dimensional random orthogonal transformations of the raw data has been proposed in Mott et al. (2020). This method is specifically suited for single-marker regression in GWAS using linear mixed models with Gaussian errors. In this thesis, we will further generalize this homomorphic encryption for genome-to-phenome analysis using mixed models.

Genome-to-phenome analyses in animal breeding often involves the estimation of genetic marker effects and breeding values, based on individual-level genotype and phenotype information. A genome-wide association study (GWAS) may also used to assess the correlations between single-nucleotide polymorphisms (SNPs) and the phenotype of interest. However, each animal breeder has a relatively small sample size, which could lead to an underpowered statistical analysis and lead to a higher chance of obtaining a false negative result. Using joint analyses by combining individual-level data before performing the analysis can increase statistical power and improve prediction accuracy, but animal breeders may be hesitant to share their animal's information with others, as this can reveal sequences responsible for their animals' economic value. One solution is to implement an encryption scheme to protect individual-level information. Homomorphic encryption for genotypes and phenotypes (HEGP) is a type of encryption that allows encrypted genomic data to be analyzed directly, providing a more secure method of estimating marker effects and breeding values when performing a joint analysis. In this study, HEGP is implemented on a real data set from a disease resilience study in pigs and evaluates the correlation between estimated marker effects and estimated breeding values of the encrypted and unencrypted data, respectively. The estimated percentages of genetic variance for each window obtained from a GWAS using the encrypted data were also compared to the results of the original study from which the data originated. Correlations between estimated marker effects, estimated breeding values, and estimated percentages of genetic variance of each window of the analyses using unencrypted data and encrypted data were all approximately 1, indicating that the implementation of HEGP in GWAS joint analyses produces effectively identical results and does not affect the precision of the obtained results.