An adaptive Bayesian polynomial chaos expansion (BPCE) is developed in this paper for uncertainty quantification (UQ) and reliability analysis. The sparsity in the PCE model is developed using automatic relevance determination (ARD) and the PCE coefficients are computed using the variational Bayesian (VB) inference. Further, Sobol sequence is utilized to evaluate a response quantity sequentially. Finally, leave one out (LOO) error is used to obtain the adaptive BPCE model. UQ and reliability analysis are performed of some numerical examples by the adaptive BPCE model. It is found that the optimal number of model evaluations and the optimal PCE degree are suitably selected simultaneously for a problem by the adaptive BPCE model. A highly accurate result is predicted by the proposed approach using very few model evaluation. Further, highly sparse PCE models are obtained by the ARD approach for most of the numerical examples. Additionally, distribution parameters of the predicted response quantity are also obtained by the VB inference, which are used to compute the confidence interval of the predicted response quantities. © 2021, The Author(s), under exclusive licence to Springer-Verlag London Ltd. part of Springer Nature.