The prevailing drive of modern seismology is to improve our knowledge of the Earth's structure, composition, and dynamics through an analysis of seismic waveforms. With increasing computing power, number and quality of seismic stations, and length of data records, the resolution and spatial coverage of current Earth models has improved substantially over the past few decades. Yet many limitations remain. The advent of ambient noise seismology has provided the solution to many issues, such as the irregular distribution of earthquakes, biases from structures outside the model region, earthquake location errors, and lack of near-surface resolution. Despite improvements to data quality and quantity and the introduction of unconventional datasets such as ambient seismic noise, a persisting shortcoming of many tomographic inversions is ad-hoc error estimation, parameterization, and regularization, which prevent a meaningful portrayal of model complexity and uncertainty. With the rapid increase in computing power, non-linear techniques based on densely sampling favorable regions of model space are now becoming tractable for real-world tomographic problems and directly address these shortcomings. One such recently introduced and promising method is transdimensional and hierarchical Bayesian inference. This alternate approach allows model parameterization and resolution to be driven by the data. This thesis presents a collection of seismic inverse problems using real world datasets, some of which are tackled using fully non-linear Bayesian statistics. The benefits of a probabilistic approach are demonstrated for datasets targeting the uppermost crust down to the core through the development of novel methods of inversion and uncertainty quantification. To begin, an unconventional methodology for studying earthquake focal mechanisms in intraplate settings is presented through the inversion of ambient noise, receiver functions, and dispersion curves. The ambient seismic noise imaging approach of this study is subsequently applied to Tasmania - to which it is highly suited - and the resulting group and phase velocity maps help decipher Tasmania's enigmatic tectonic history. The same ambient noise dataset is further manipulated to yield a 3D shear velocity model of the region using a two-step transdimensional, hierarchical ensemble inference approach. Two prominent low-velocity anomalies offer insight into the Paleozoic evolution of the east Gondwana margin and support a connection between Tasmania and mainland Australia since the Cambrian. This approach is also applied to a larger dataset encompassing much of mainland southeast Australia. The Bayesian approach is also applied to a global dataset of differential body wave travel times in an effort to reveal P-wave velocity heterogeneity in the lowermost mantle. Another deep Earth application is demonstrated through an inversion for the time-dependent differential rotation of the inner core with respect to the rest of the mantle using careful measurements of earthquake doublets. The transdimensional nature of the inversion problem means that the data drive the number of free parameters constraining the differential rotation pattern, which exhibits much more complexity than the simple linear trend long-promoted by previous studies. The contents of this thesis help augment the diverse and wide-reaching applications for Bayesian statistics, which will continue to improve with future increases in computational power
