Full-sampling Magnetic Resonance Imaging （MRI） requires a long scan time， which not only limits the examination efficiency， but also tends to induce motion artifacts due to subject movement. To address the issues of high parameter sensitivity and inability to quantify result uncertainty in traditional Compressed Sensing MRI （CS-MRI） reconstruction methods， an uncertainty-aware MRI reconstruction method based on wavelet-domain Sparse Bayesian Learning （SBL） was proposed， namely BU-MRI （Bayesian Uncertainty-guided MRI）. Firstly， the advantage of the wavelet transform in multi-resolution image representation was leveraged， and a hierarchical Bayesian probability model was constructed by characterizing the sparsity of MRI images in the wavelet domain as a prior. Secondly， a posterior inference strategy combining Gibbs sampling and marginal likelihood maximization was adopted to achieve effective estimation of high-dimensional sparse coefficients and adaptive hyperparameters updating. Finally， based on the updated model parameters， high-quality images were iteratively reconstructed from undersampled K-space data. Furthermore， pixel-wise posterior confidence intervals could be provided， offering a quantitative assessment of uncertainty in the reconstruction results. Experimental results on both simulated and real MRI data demonstrated that BU-MRI outperformed methods such as Zero-Filled Inverse Discrete Fourier Transform （ZF-IDFT） and k-t Robust Principal Component Analysis （k-t RPCA） in terms of Peak Signal-to-Noise Ratio （PSNR） and Structural Similarity Index Measure （SSIM）. On real cardiac MRI data and brain MRI data with a sampling rate of 0.5， the PSNR of BU-MRI achieved 44.42 dB and 40.37 dB， and SSIM reached 0.976 5 and 0.954 7， respectively. BU-MRI exhibits excellent performance in structural fidelity， error suppression， and frequency-domain consistency. It shows stable convergence and robustness across various sampling rates and noise levels， providing a reliable reconstruction framework with uncertainty quantification capability for clinical MRI.