The prior information of channel can't be induced in the process of channel estimation in MIMO relay communication system. To solve the problem

a novel semi-blind channel estimation based on variational inference is proposed. In this algorithm

some latent hyper-parameters such as factor precision

noise precision are introduced into the algorithm

and channel estimation probability model is built based on nested PARAFAC tensor decomposition. Since the posterior probability distribution of the channel parameters is complex

some point estimation methods

such as traditional maximum likelihood and maximum posteriori algorithm

are difficult to implement. The iteration formulas of factor matrix

factor precision and noise precision are deduced by the idea of variational inference principle

making the q distribution

which has the factor decomposition form

approach the unknown parameter posterior distribution. In addition

low bound of model evidence

model initiation and algorithm complexity are also analyzed. The algorithm can utilize the prior information of channel to improve channel estimation performance. The parameters can be tuned automatically

and complexity is linear with the dimension of observed data. Simulations show that the proposed algorithm has better estimation performance under flat Rayleigh channel condition

compared with No-blind algorithm

Alternating Least Square (ALS) based algorithm and No-linear Least Square (NLS) based algorithm

and has lower complexity and faster convergence speed

compared with alternating least square algorithm.