This paper focuses on patrolling missions for monitoring visitors in an environment. A single mobile robot is used. For the robot, it is required to monitor as many visitors as possible utilizing the mobility and monitoring capabilities. A challenge in the patrolling missions is that the robot does not have information about the visitors beforehand. For the unknown visitors, the robot is required to identify the trends. For this purpose, a Bayesian learning approach is applied to this robot. The identified visitor trends allow the robot to employ patrolling strategies. This is the adaptability of the patrolling robot to the unknown visitors. In this paper, we present an optimal patrolling strategy based on a value iteration method. The robot is enabled to have an optimal policy that suggests three patrolling decisions, such as the target monitoring area, monitoring time, and patrolling path in an integrated way. Furthermore, the optimal patrolling strategy is improved from the viewpoint of exploration and exploitation of information about the unknown visitors in the environment. Through simulation experiments, we discuss the effectiveness of the patrolling strategy.