I provide a very brief synopsis of my investigations into the intersections between differential geometry and Bayesian analysis. My thesis combines these two disciplines with the hope that a synergy might emerge and facilitate the useful application of Bayesian inference to real-world science. In particular, dynamic and high-dimensional neural data provides a challenging litmus test for the proposed methods.
A major component of my work is the development and application of probabilistic models defined over smooth manifolds: dependencies between time series are modeled using the manifold of Hermitian positive definite matrices; probability density functions are modeled using the infinite sphere; and high-dimensional data are modeled using the Stiefel manifold. Whereas formulating a manifold-based model is not difficult—in a certain sense, the geometry occurs a priori in each of the cases considered—the non-trivial geometry presents computational challenges for model-based inference. Hence, my dissertation contributes two new algorithms for Bayesian inference on Riemannian manifolds. In turn, these algorithms power Bayesian neural decoding and spectral inference.