Changepoint detection and time series analysis are vital for applications across finance, biomedicine, and sociology. Existing methods often assume uniform sampling or simplified data-generating mechanisms, and typically return a single set of changepoints, offering limited insight into uncertainty. This dissertation introduces flexible Bayesian and generative frameworks to overcome these limitations. In Section 1, we propose a Bayesian approach that treats partitions as random objects, enabling principled characterization of both changepoint presence and location uncertainty. Using a depth-based geometry for posterior partition distributions, we show how this resolves issues with thresholding MCMC draws, which fail to capture partition ordering or variability in changepoint locations. Simulations demonstrate improved detection of subtle changes, clearer separation of presence versus location uncertainty, and more stable inference compared to classical methods. Section 2 extends adaptive Bayesian trend filtering to irregularly sampled time series, accommodating heterogeneous temporal resolution and enabling realistic modeling of complex signals such as heart rate dynamics. Finally, in Section 3, we develop a generative-adversarial approach for fair classification that balances predictive performance with mitigation of group-level biases through the synthesis of realistic, demographically balanced training data.