  Concept bottleneck model (CBM) is a ubiquitous method that can interpret
neural networks using concepts. In CBM, concepts are inserted between the
output layer and the last intermediate layer as observable values. This helps
in understanding the reason behind the outputs generated by the neural
networks: the weights corresponding to the concepts from the last hidden layer
to the output layer. However, it has not yet been possible to understand the
behavior of the generalization error in CBM since a neural network is a
singular statistical model in general. When the model is singular, a one to one
map from the parameters to probability distributions cannot be created. This
non-identifiability makes it difficult to analyze the generalization
performance. In this study, we mathematically clarify the Bayesian
generalization error and free energy of CBM when its architecture is
three-layered linear neural networks. We also consider a multitask problem
where the neural network outputs not only the original output but also the
concepts. The results show that CBM drastically changes the behavior of the
parameter region and the Bayesian generalization error in three-layered linear
neural networks as compared with the standard version, whereas the multitask
formulation does not.
