A Bayes optimal predicting method of Bayesian network with visualization of causal relationship

A Bayesian network is one of the useful models for pattern recognition problems and it has the features of both stochastic prediction and causal models. A Bayesian network expresses the causal relationship between variables with directed graphs. Usually the structure of a Bayesian network is statistically estimated using a set of training data and the model selection has been applied in conventional methods when Bayesian network structures were estimated. However, it is not necessary to choose one model for the purpose of prediction. From the viewpoint of Bayesian statistics, it is well known that prediction using the mixture model on model class is Bayes optimal. In general, the mixture model that is given by a weighted sum of all models with the posterior probability on the model class is the Bayes optimal prediction. In this paper, we propose an new Bayes optimal prediction on a Bayesian network model class using the mixture model. A mixture model sometimes becomes a complex expression due to the weighted sum of all models on a model class, and it results in loss of the usefulness as a causal model. Since the easiness of interpretation is one of the merits of a Bayesian network, using a mixed model only for improvement in predictive accuracy may lead to losing the merit of a Bayesian network. Therefore, we propose a new method that is configured with the mixture model utilizing the characteristics of the Bayesian network by organizing model classes properly. Furthermore, we propose a method to quantitatively assess the strength of the causal relationship between the nodes on the mixed Bayesian network model. In addition, the effectiveness of the proposed method is clarified via a numerical experiment on an application to a prediction problem of buying and selling of shares stock market.