Concept Bottleneck Model (CBM) is a methods for explaining neural networks. In CBM, concepts which correspond to reasons of outputs are inserted in the last intermediate layer as observed values. It is expected that we can interpret the relationship between the output and concept similar to linear regression. However, this interpretation requires observing all concepts and decreases the generalization performance of neural networks. Partial CBM (PCBM), which uses partially observed concepts, has been devised to resolve these difficulties. Although some numerical experiments suggest that the generalization performance of PCBMs is almost as high as that of the original neural networks, the theoretical behavior of its generalization error has not been yet clarified since PCBM is singular statistical model. In this paper, we reveal the Bayesian generalization error in PCBM with a three-layered and linear architecture. The result indcates that the structure of partially observed concepts decreases the Bayesian generalization error compared with that of CBM (full-observed concepts).

翻译：概念瓶颈模型（CBM）是一种解释神经网络的方法。在该模型中，对应于输出结果原因的概念被作为观测值插入到最后一个中间层。我们期望能像线性回归一样解释输出与概念之间的关系。然而，这种解释需要观测所有概念，从而降低了神经网络的泛化性能。部分CBM（PCBM）通过使用部分观测概念来解决这些困难。尽管一些数值实验表明PCBM的泛化性能几乎与原始神经网络相当，但由于PCBM属于奇异统计模型，其泛化误差的理论行为尚未明确。本文揭示了具有三层线性架构的PCBM中贝叶斯泛化误差的规律。结果表明，与CBM（全观测概念）相比，部分观测概念的结构降低了贝叶斯泛化误差。