Anxiety and depression are highly prevalent diseases with common heritable elements, but the particular genetic mechanisms and biological pathways underlying them are poorly understood. Part of the challenge in understanding the genetic basis of these disorders is that they are polygenic and often context-dependent. In my thesis, I apply a series of modern statistical tools to ascertain some of the myriad genetic and environmental factors that underlie fear-related behaviours in nearly two thousand heterogeneous stock mice, which serve as animal models of anxiety and depression. Using a Bayesian method called Sparse Partitioning and a frequentist method called Bagphenotype, I identify gene-by-sex interactions that contribute to variation in fear-related behaviours, such as those displayed in the elevated plus maze and the open field test, although I demonstrate that the contributions are generally small. Also using Bagphenotype, I identify hundreds of gene-by-environment interactions related to these traits. The interacting environmental covariates are diverse, ranging from experimenter to season of the year. With gene expression data from a brain structure associated with anxiety called the hippocampus, I generate modules of co-expressed genes and map them to the genome. Two of these modules were enriched for key nervous system components — one for dendritic spines, another for oligodendrocyte markers — but I was unable to find significant correlations between them and fear-related behaviours. Finally, I employed another Bayesian technique, Sparse Instrumental Variables, which takes advantage of conditional probabilities to identify hippocampus genes whose expression appears not just to be associated with variation in fear-related behaviours, but cause variation in those phenotypes.

<p>Graphs are a natural choice for understanding the associations between variables, and assuming a probabilistic embedding for the graph structure leads to a variety of graphical models that enable us to understand these associations even further. In the realm of high-dimensional data, where the number of associations between interacting variables is far greater than the available number of data points, the goal is to infer a sparse graph. In this thesis, we make contributions in the domain of Bayesian graphical models, where our prior belief on the graph structure, encoded via uncertainty on the model parameters, enables the estimation of sparse graphs.</p> <p><br></p> <p>We begin with the Gaussian Graphical Model (GGM) in Chapter 2, one of the simplest and most famous graphical models, where the joint distribution of interacting variables is assumed to be Gaussian. In GGMs, the conditional independence among variables is encoded in the inverse of the covariance matrix, also known as the precision matrix. Under a Bayesian framework, we propose a novel prior--penalty dual called the `graphical horseshoe-like' prior and penalty, to estimate precision matrix. We also establish the posterior convergence of the precision matrix estimate and the frequentist consistency of the maximum a posteriori (MAP) estimator.</p> <p><br></p> <p>In Chapter 3, we develop a general framework based on local linear approximation for MAP estimation of the precision matrix in GGMs. This general framework holds true for any graphical prior, where the element-wise priors can be written as a Laplace scale mixture. As an application of the framework, we perform MAP estimation of the precision matrix under the graphical horseshoe penalty.</p> <p><br></p> <p>In Chapter 4, we focus on graphical models where the joint distribution of interacting variables cannot be assumed Gaussian. Motivated by the quantile graphical models, where the Gaussian likelihood assumption is relaxed, we draw inspiration from the domain of precision medicine, where personalized inference is crucial to tailor individual-specific treatment plans. With an aim to infer Directed Acyclic Graphs (DAGs), we propose a novel quantile DAG learning framework, where the DAGs depend on individual-specific covariates, making personalized inference possible. We demonstrate the potential of this framework in the regime of precision medicine by applying it to infer protein-protein interaction networks in Lung adenocarcinoma and Lung squamous cell carcinoma.</p> <p><br></p> <p>Finally, we conclude this thesis in Chapter 5, by developing a novel framework to compute the marginal likelihood in a GGM, addressing a longstanding open problem. Under this framework, we can compute the marginal likelihood for a broad class of priors on the precision matrix, where the element-wise priors on the diagonal entries can be written as gamma or scale mixtures of gamma random variables and those on the off-diagonal terms can be represented as normal or scale mixtures of normal. This result paves new roads for model selection using Bayes factors and tuning of prior hyper-parameters.</p>