  Bayesian learning is a powerful learning framework which combines the
external information of the data (background information) with the internal
information (training data) in a logically consistent way in inference and
prediction. By Bayes rule, the external information (prior distribution) and
the internal information (training data likelihood) are combined coherently,
and the posterior distribution and the posterior predictive (marginal)
distribution obtained by Bayes rule summarize the total information needed in
the inference and prediction, respectively. In this paper, we study the
Bayesian framework of the Tensor Network from two perspective. First, we
introduce the prior distribution to the weights in the Tensor Network and
predict the labels of the new observations by the posterior predictive
(marginal) distribution. Since the intractability of the parameter integral in
the normalization constant computation, we approximate the posterior predictive
distribution by Laplace approximation and obtain the out-product approximation
of the hessian matrix of the posterior distribution of the Tensor Network
model. Second, to estimate the parameters of the stationary mode, we propose a
stable initialization trick to accelerate the inference process by which the
Tensor Network can converge to the stationary path more efficiently and stably
with gradient descent method. We verify our work on the MNIST, Phishing Website
and Breast Cancer data set. We study the Bayesian properties of the Bayesian
Tensor Network by visualizing the parameters of the model and the decision
boundaries in the two dimensional synthetic data set. For a application
purpose, our work can reduce the overfitting and improve the performance of
normal Tensor Network model.
