This article proposes a Bayesian learning framework for online change detection of high-dimensional data streams where only a subset of variables can be observed at each time point due to limited sensing capacities. On the one hand, we need to build a change detection scheme based on partial observations. On the other, the scheme should be able to adaptively and actively select the most critical sensing variables to observe to maximize the detection power. To address these two points, in this article, first, a novel Bayesian Spike-Slab Composite Decomposition (BSSCD) is proposed to decompose the high-dimensional signals onto normal and abnormal bases, where the projection coefficients are efficiently estimated via variational Bayesian inference. Built upon it, the posterior Bayes factor is constructed as the detection statistic. Second, by further formulating the detection statistic as the reward function of combinatorial multi-armed bandit (CMAB), a Thompson sampling strategy is proposed for selecting the potential changed variables with the balance of exploration and exploitation. The efficacy and applicability of our method are demonstrated in practice with numerical studies and a real case study.