This paper proposes a deep neural network named BayesianPUFNet that can achieve high prediction accuracy even with few challenge-response pairs (CRPs) available for training. Generally, modeling attacks are a vulnerability that could compromise the authenticity of physically unclonable functions (PUFs); thus, various machine learning methods including deep neural networks have been proposed to assess the vulnerability of PUFs. However, conventional modeling attacks have not considered the cost of CRP collection and analyzed attacks based on the assumption that sufficient CRPs were available for training; therefore, previous studies may have underestimated the vulnerability of PUFs. Herein, we show that the application of Bayesian deep neural networks that incorporate Bayesian statistics can provide accurate response prediction even in situations where sufficient CRPs are not available for learning. Numerical experiments show that the proposed model uses only half the CRP to achieve the same response prediction as that of the conventional methods. Our code is openly available on https://github.com/bayesian-puf-net/bayesian-puf-net.git.