Bayesian optimization (BO) is an approach for optimizing an expensive-to-evaluate black-box function and sequentially determines values of input variables to evaluate the function. However, it is expensive or difficult to specify values of all input variables, for example, in outsourcing scenarios where production of input queries with many input variables involves much cost. In this paper, we propose a novel Gaussian process bandit problem, BO with partially specified queries (BOPSQ). In BOPSQ, unlike the standard BO setting, a learner specifies only the values of some input variables, and the values of the unspecified input variables are randomly determined according to a known or unknown distribution. We propose two algorithms based on posterior sampling for cases of known and unknown input distributions. We further derive their regret bounds that are sublinear for popular kernels. We demonstrate the effectiveness of the proposed algorithms using test functions and real-world datasets.