  Accurately determining the effective Hubbard parameter $(U_{eff})$ in Density
Functional Theory plus U (DFT+U) remains a significant challenge, often relying
on empirical methods or linear response theory, which frequently fail to
predict accurate material properties. This study introduces BMach, an advanced
Bayesian optimization algorithm that refines $U_{eff}$ by incorporating
electronic properties, such as band gaps and eigenvalues, alongside structural
properties like lattice parameters. Implemented within the Quantum Espresso
platform, BMach demonstrates superior accuracy and reduced computational cost
compared to traditional methods. The BMach-optimized $U_{eff}$ values yield
electronic properties that align closely with experimental and high-level
theoretical results, providing a robust framework for high-throughput materials
discovery and detailed electronic property characterization across diverse
material systems.
