Statistical learning posits a structured relationship between observed data and unobserved quantities — latent components underlying a mixture, counterfactual outcomes unobserved under the realized treatment assignment, or low-dimensional representations encoding complex generative factors — and makes inference over the parameters that govern this relationship. In each case, a graphical model encodes the structural assumptions through conditional independence and factorization, but inference over the resulting distributional objects demands tools that are stable under the geometric irregularities — limited overlap, high dimensionality, unknown model complexity — that arise in practice. This thesis develops a distributional framework that pairs graphical model structure with optimal transport geometry to address this need. We apply the framework to causal inference under limited overlap, replacing density-ratio reweighting with geometrically stable transport maps and developing Wasserstein-based sensitivity analysis for partial identification; to structured generative modeling, introducing Structured Flow Autoencoders that combine conditional normalizing flows with latent graphical models via a novel flow matching objective; and to mixture model estimation, where Bayes fixed-point iteration and entropy-regularized semi-discrete optimal transport yield a geometry-driven approach to component recovery and model selection. Across all settings, the thesis demonstrates that replacing pointwise inference procedures with distributional, geometry-aware ones — anchored by graphical model structure — yields methods that are simultaneously more principled and more practically reliable.