Obtaining the channel state information (CSI) is a challenging problem in reconfigurable intelligent surfaces (RIS)-assisted wireless communication systems due to the passive nature of the RIS elements. In this thesis, we study a variational inference (VI)-based CSI estimation approach in a fully passive RIS-aided mmWave single-user single-input multiple-output (SIMO) communication system. Specifically, first, we introduce the VI-based estimation framework in a communication system along with the neural variational networks used to estimate the CSI in a communication system based on the training signals. Then, we propose a VI-based joint channel estimation method to estimate the user-equipment (UE)-to-RIS (UE-RIS) and RIS-to-base station (RIS-BS) channels using uplink training signals in a passive RIS setup. However, updating the phase-shifts based on the instantaneous CSI (I-CSI) leads to a high signaling overhead especially due to the short coherence block of the UE-RIS channel. Therefore, to reduce the signaling complexity for updating the phase-shifts, we propose a VI-based method to estimate the RIS-BS channel along with the covariance matrix of the UE-RIS channel that remains quasi-static for a longer period than the instantaneous UE-RIS channel. In the VI framework, we approximate the posterior of the channel gains/channel covariance matrix with convenient distributions given the received uplink training signals. The parameters of the approximated distributions are generated by deep neural networks trained using variational loss functions derived using the lower bound on the log-likelihood of the received signal. Then, the learned distributions, which are close to the true posterior distributions in terms of Kullback-Leibler divergence, are leveraged to obtain the maximum a posteriori (MAP) estimation of the considered CSI. The simulation results demonstrate that MAP channel estimation using approximated posteriors yields a capacity that is close to the one achieved with true posteriors, thus demonstrating the effectiveness of the proposed methods. Furthermore, our results show that estimating the channel covariance matrix improves the spectral efficiency by reducing the pilot signaling required to obtain the phase-shifts for the RIS elements in a channel-varying environment.