Abstract How can we decompose a data tensor if the indices are partially missing? Tensor decomposition is a fundamental tool to analyze the tensor data. Suppose, for example, we have a 3rd-order tensor X where each element Xijk takes 1 if user i posts word j at location k on Twitter. Standard tensor decomposition expects all the indices are observed. However, in some tweets, location k can be missing. In this paper, we study a tensor decomposition problem where the indices (i, j, or k) of some observed elements are partially missing. Towards the problem, we propose a probabilistic tensor decomposition model that handles missing indices as latent variables. To infer them, we develop an algorithm based on the variational MAP-EM algorithm, which enables us to leverage the information from the incomplete data. The experiments on both synthetic and real datasets show that the proposed model achieves higher accuracy in the tensor completion task than baselines.