Recent advancements in analyzing Pulsar Timing Arrays (PTA) data focus on improving the measurement of low-frequency gravitational waves, a key goal in astrophysics research. A significant challenge in this domain arises from complex noise processes that can obscure the signals of interest. To address this, researchers have proposed a hierarchical Bayesian modeling approach that aims to enhance the accuracy and robustness of noise characterization and stochastic gravitational wave background (SGWB) inference. This technique involves parameter decorrelation to mitigate prior dependence, thereby refining the extraction of meaningful signals from noisy data. By systematically accounting for noise and signal uncertainties within a unified framework, the method improves confidence in gravitational wave detection efforts. These developments represent a critical step forward in PTA data analysis, offering a more reliable pathway to understanding the low-frequency gravitational wave universe. The approach aligns with ongoing efforts to overcome analytical challenges and achieve precise astrophysical measurements.