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Quantum amplitude estimation is a fundamental routine that offers a quadratic speed-up over classical approaches. The original QAE protocol is based on phase estimation. The associated circuit depth and width, and the assumptions of fault tolerance, are unfavorable for near-term quantum technology. Subsequent approaches attempt to replace the original protocol with hybrid iterative quantum-classical strategies. In this work, we introduce BAE, a noise-aware Bayesian algorithm for QAE that combines quantum circuits with a statistical inference backbone. BAE can dynamically characterize device noise and adapt to it in real-time. Problem-specific insights and approximations are used to keep the problem tractable. We further propose an annealed variant of BAE, drawing on methods from statistical inference, to enhance statistical robustness. Our proposal is parallelizable in both quantum and classical components, offers tools for fast noise model assessment, and can leverage preexisting information. Additionally, it accommodates experimental limitations and preferred cost trade-offs. We show that BAE achieves Heisenberg-limited estimation and benchmark it against other approaches, demonstrating its competitive performance in both noisy and noiseless scenarios.