This paper presents a new method for reliability analysis with partially observed information, which integrates the Bayesian Gaussian process latent variable model (BGP-LVM) with adaptive sampling to iteratively select new partially observable training sample points to improve the reliability modeling efficiency. BGP-LVM is used to exploit all available information, instead of using the fully observed part only. Then, a new adaptive sampling criterion is designed for partially observed information considering the missing frame and information cost and used to iteratively select new training sample points to further improve the efficiency of the surrogate model development process. To the best of the authors' knowledge, this is the first work designing adaptive sampling and adaptive surrogate modeling approaches for the dataset containing missing values. The numerical experiments demonstrated that the proposed adaptive surrogate modeling method can effectively utilize all available information including fully observed data and partially observed data. More importantly, it provides a more accurate and cost-efficient result by taking advantage of extra partially observed information.