  Accurate calibration of control parameters in quantum gates is crucial for
high-fidelity operations, yet it represents a significant time and resource
challenge, necessitating periods of downtime for quantum computers. Robust
Phase Estimation (RPE) has emerged as a practical and effective calibration
technique aimed at tackling this challenge. It combines a provably efficient
number of control pulses with a classical post-processing algorithm to estimate
the phase accumulated by a quantum gate. We introduce Bayesian Robust Phase
Estimation (BRPE), an innovative approach that integrates Bayesian parameter
estimation into the classical post-processing phase to reduce the sampling
overhead. Our numerical analysis shows that BRPE markedly reduces phase
estimation errors, requiring approximately $50\%$ fewer samples than standard
RPE. Specifically, in an ideal, noise-free setting, it achieves up to a $96\%$
reduction in average absolute estimation error for a fixed sample cost of $88$
shots when compared to RPE. Under a depolarizing noise model, it attains up to
a $47\%$ reduction for a fixed cost of $176$ shots. Additionally, we adapt BRPE
for Ramsey spectroscopy applications and successfully implement it
experimentally in a trapped ion system.
