  For a broad variety of critical applications, it is essential to know how
confident a classification prediction is. In this paper, we discuss the
drawbacks of softmax to calculate class probabilities and to handle uncertainty
in Bayesian neural networks. We introduce a new kind of prediction layer called
radial prediction layer (RPL) to overcome these issues. In contrast to the
softmax classification, RPL is based on the open-world assumption. Therefore,
the class prediction probabilities are much more meaningful to assess the
uncertainty concerning the novelty of the input. We show that neural networks
with RPLs can be learned in the same way as neural networks using softmax. On a
2D toy data set (spiral data), we demonstrate the fundamental principles and
advantages. On the real-world ImageNet data set, we show that the open-world
properties are beneficially fulfilled. Additionally, we show that RPLs are less
sensible to adversarial attacks on the MNIST data set. Due to its features, we
expect RPL to be beneficial in a broad variety of applications, especially in
critical environments, such as medicine or autonomous driving.
