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In this paper, we investigate the content popularity prediction problem in cache-enabled fog radio access networks (F-RANs). In order to predict the content popularity with high accuracy and low complexity, we propose a Gaussian process based regressor to model the content request pattern. Firstly, the relationship between content features and popularity is captured by our proposed model. Then, we utilize Bayesian learning to train the model parameters, which is robust to overfitting. However, Bayesian methods are usually unable to find a closed-form expression of the posterior distribution. To tackle this issue, we apply a stochastic variance reduced gradient Hamiltonian Monte Carlo (SVRG-HMC) method to approximate the posterior distribution. To utilize the computing resources of other fog access points (F-APs) and to reduce the communications overhead, we propose a quantized federated learning (FL) framework combining with Bayesian learning. The quantized federated Bayesian learning framework allows each F-AP to send gradients to the cloud server after quantizing and encoding. It can achieve a tradeoff between prediction accuracy and communications overhead effectively. Simulation results show that the performance of our proposed policy outperforms the existing policies.