In this paper, we first analyze the discriminative power in the Best Match (BM) 25 formula and provide its calculation method from the Bayesian point of view. The resulting, derived discriminative power is quite similar to the exponential inverse document frequency (EIDF) that we have previously proposed [1] but retains more preferable theoretical advantages. In our previous paper [1], we proposed the EIDF in the framework of the probabilistic information retrieval (IR) method BM25 to address the instance search task, which is a specific object search for videos using an image query. Although the effectiveness of our EIDF was experimentally demonstrated, we did not consider its theoretical justification and interpretation. We also did not describe the use of region-of-interest (ROI) information, which is supposed to be input to the instance search system together with the original image query showing the instance. Therefore, here, we justify the EIDF by calculating the discriminative power in the BM25 from the Bayesian viewpoint. We also investigate the effect of the ROI information for improving the instance search accuracy and propose two search methods incorporating the ROI effect into the BM25 video ranking function. We validated the proposed methods through a series of experiments using the TREC Video Retrieval Evaluation instance search task dataset.