Edge-caching is recognized as an efficient technique for future cellular networks to improve network capacity and user-perceived quality of experience. To enhance the performance of caching systems, designing an accurate content request prediction algorithm plays an important role. In this paper, we develop a flexible model, a Poisson regressor based on a Gaussian process, for the content request distribution. The first important advantage of the proposed model is that it encourages the already existing or seen contents with similar features to be correlated in the feature space and therefore it acts as a regularizer for the estimation. Second, it allows to predict the popularities of newly-added or unseen contents whose statistical data is not available in advance. In order to learn the model parameters, which yield the Poisson arrival rates or alternatively the content \textit{popularities}, we invoke the Bayesian approach which is robust against over-fitting. However, the resulting posterior distribution is analytically intractable to compute. To tackle this, we apply a Markov Chain Monte Carlo (MCMC) method to approximate this distribution which is also asymptotically exact. Nevertheless, the MCMC is computationally demanding especially when the number of contents is large. Thus, we employ the Variational Bayes (VB) method as an alternative low complexity solution. More specifically, the VB method addresses the approximation of the posterior distribution through an optimization problem. Subsequently, we present a fast block-coordinate descent algorithm to solve this optimization problem. Finally, extensive simulation results both on synthetic and real-world datasets are provided to show the accuracy of our prediction algorithm and the cache hit ratio (CHR) gain compared to existing methods from the literature.