Device activity detection is one main challenge in grant-free random access, which is recently proposed to support massive access for massive machine-type communications (mMTC). Existing solutions fail to consider interference generated by massive Internet of Things (IoT) devices, or important prior information on device activities and interference. In this paper, we consider device activity detection at an access point (AP) in the presence of interference generated by massive devices from other cells. We consider the joint maximum likelihood (ML) estimation and the joint maximum a posterior probability (MAP) estimation of both the device activities and interference powers, jointly utilizing tools from probability, stochastic geometry and optimization. Each estimation problem is a difference of convex (DC) programming problem, and a coordinate descent algorithm is proposed to obtain a stationary point. The proposed ML estimation extends the existing ML estimation by considering the estimation of interference powers together with the estimation of device activities. The proposed MAP estimation further enhances the proposed ML estimation by exploiting prior distributions of device activities and interference powers. Numerical results show the substantial gains of the proposed joint estimation designs, and reveal the importance of explicit consideration of interference and the value of prior information in device activity detection.