The problem of obtaining spectral densities from lattice data has been receiving great attention due to its importance in understanding many aspects of the Standard Model and beyond. Due to the challenging nature of the problem, different methods have been devised in order to improve our ability to provide stable and reliable solutions. In this talk, we review two seemingly different approaches: Backus-Gilbert type of solutions, and Bayesian methods based on Gaussian Processes. Despite the different underlying philosophies, these frameworks show a striking amount of similarities. After showing how an exact relation between these two methods can be drawn, we shall give a fully Bayesian formulation of our natively frequentist approach proposed with M. Hansen and N. Tantalo. We shall discuss the benefits of this dual formulation and review some recent applications.