The computation-intensive circuit simulation makes the analog circuit sizing challenging for large-scale/complicated analog/RF circuits. A Bayesian optimization approach has been proposed recently for the optimization problems involving the evaluations of black-box functions with high computational cost in either objective functions or constraints. In this paper, we propose a weighted expected improvement-based Bayesian optimization approach for automated analog circuit sizing. Gaussian processes (GP) are used as the online surrogate models for circuit performances. Expected improvement is selected as the acquisition function to balance the exploration and exploitation during the optimization procedure. The expected improvement is weighted by the probability of satisfying the constraints. In this paper, we propose a complete Bayesian optimization framework for the optimization of analog circuits with constraints for the first time. The existing GP model-based optimization methods for analog circuits take the GP models as either offline models or as assistance for the evolutionary algorithms. We also extend the Bayesian optimization algorithm to handle multi-objective optimization problems. Compared with the state-of-the-art approaches listed in this paper, the proposed Bayesian optimization method achieves better optimization results with significantly less number of simulations.