When it comes to probability theory we all would have heard of joint distribution, marginal distribution, independence etc. In this article I will focus my attention onto independence specially conditional independence. In others words if the happening of event A doesn't affect the probability of event B happening, both events are said to be independent. From the view of information theory it can be interpreted as: if knowing A doesn't provide any additional information about B, then A and B are said to be independent. These are the different interpretations for the concept of independence.

Probability is one of the most common terminologies, not only in mathematics but also in the real world. We use the word probability frequently. About seven years ago, I was in my secondary level of education and got introduced to the term probability as a topic of mathematics. At that time, I had solved so many mathematical problems regarding probability. Unfortunately, it did not seem interesting to me.

Describing or calculating the conditional probabilities of multiple events is exponentially expensive. In this work, a natural generalization of Bayesian belief network is proposed by incorporating with tensor network, which is dubbed as Bayesian tensor network (BTN), to efficiently describe the conditional probabilities among multiple sets of events. The complexity of BTN that gives the conditional probabilities of $M$ sets of events scales only polynomially with $M$. To testify its validity, BTN is implemented to capture the conditional probabilities between images and their classifications, where each feature is mapped to a probability distribution of a set of mutually exclusive events. A rotation optimization method is suggested to update BTN, which avoids gradient vanishing problem and exhibits high efficiency. With a simple tree network structures, BTN exhibits competitive performances on fashion-MNIST dataset. Analogous to the tensor network simulations of quantum systems, the validity of BTN implies an "area law" of fluctuations in image recognition problems.