## 1 Introduction
### Previous Research and Limitations
[24] proposed a deep reinforcement learning algorithm for double coded caching. In summary, DL-based CE algorithms use deep neural networks to learn the channel feature, and obtain the complete channel information from received pilot symbols. Some DL-based CE models treat channel information as a picture and use the CS-based method to solve channel state information. Despite the high accuracy of DL-based CE algorithms, it is impractical to obtain labeled data or train self-adaptive neural networks for channel estimation. Therefore, it is still difficult to develop DL-based CE algorithms that have real-time generality and efficiency properties. Some researchers try to use statistical methods to solve the CE problem by forming it into a statistical optimization problem. Treating the channel state information as a set of samples generated by a distribution, we can write the joint posterior density to be maximized as the objective function. However, in statistical inference, it is often too difficult to directly optimize the objective function when computing joint posterior density or other statistical variables. Therefore, we opt for an alternative suboptimal approach, which is VI. In VI, we use another distribution that has minimal KL divergence, or variational free energy, or evidence lower bound (ELBO), with the approximated distribution. By approximating the difficult-to-compute distribution, we can guarantee faster convergence and optimum performance (see [PERSON] et al. [25]). Please note that many studies that use MP, MF and BP are categorized as VI, and use factor graphs as their graphical models. We classify MP-based channel estimation (CE) algorithms as a specific group, because unlike MF and BP, MP is the direct model for autonomous systems that communicate two-way. In 2008, [PERSON] et al. [26] proposed a divergence minimization approach to derive iterative decoding algorithms in coded CDMA systems. They stated that the KL divergence is equivalent to the variational free energy, both of which can be used to measure the distribution distance between true distribution and auxiliary distribution. In 2010, [PERSON] et al. [27] proposed a variational message passing (VMP) architecture that can estimate the channel coefficients and inverse noise variance matrix in a semi-blind way. As with [PERSON] et al. [26], this scheme is also an iterative decoding channel estimation approach, which only differs in proposing the notion of VI. In 2012, [PERSON] et al. [25] proposed a simpler suboptimal approach that uses VI and obtains the auxiliary distribution through KL divergence minimizing and joint posterior density maximizing (we call joint distributed CE in the following). In 2019, [PERSON] et al. [21] proposed a novel Bayes-optimal channel estimation method that makes use of approximate belief propagation Bayesian networks. In 2017, [PERSON] et al. [28] proposed a different variational Bayesian inference-basedchannel estimation algorithm using a Gaussian mixture as a prior distribution. By using this Gaussian mixture in the prior distribution, the proposed algorithm can make the best use of sparsity within a single channel matrix, as well as between different channel matrices. In 2019, [PERSON] et al. [29] proposed another variational Bayesian channel estimation algorithm to estimate channel coefficients and the hyper-parameter of the transmitted symbol in MIMO cases. They modeled the MIMO system as a directed graphical model and so, despite the advantages offered by VI combined with data subsampling, i.e., providing approximate posterior inference over large data sets, their approach suffers from a poor local optimum (see [PERSON] et al. [30]). Some researchers use an annealing approach to tackle this problem, but annealing is only effective when choosing the appropriate cooling schedule (see [PERSON] et al. [30]).