In this paper, we study pseudo-labelling. Pseudo-labelling employs raw inferences on unlabelled data as pseudo-labels for self-training. We elucidate the empirical successes of pseudo-labelling by establishing a link between this technique and the Expectation Maximisation algorithm. Through this, we realise that the original pseudo-labelling serves as an empirical estimation of its more comprehensive underlying formulation. Following this insight, we present a full generalisation of pseudo-labels under Bayes’ theorem, termed Bayesian Pseudo Labels. Subsequently, we introduce a variational approach to generate these Bayesian Pseudo Labels, involving the learning of a threshold to automatically select high-quality pseudo labels. In the remainder of the paper, we showcase the applications of pseudo-labelling and its generalised form, Bayesian Pseudo-Labelling, in the semi-supervised segmentation of medical images. Specifically, we focus on: (1) 3D binary segmentation of lung vessels from CT volumes; (2) 2D multi-class segmentation of brain tumours from MRI volumes; (3) 3D binary segmentation of whole brain tumours from MRI volumes; and (4) 3D binary segmentation of prostate from MRI volumes. We further demonstrate that pseudo-labels can enhance the robustness of the learned representations. The code is released in the following GitHub repository: .

中文翻译：

---

期望最大化伪标签

在本文中，我们研究伪标签。伪标签利用对未标记数据的原始推断作为自我训练的伪标签。我们通过在伪标签技术和期望最大化算法之间建立联系来阐明伪标签的经验成功。通过这一点，我们意识到原始的伪标签是对其更全面的基础公式的经验估计。根据这一见解，我们提出了贝叶斯定理下伪标签的完整概括，称为贝叶斯伪标签。随后，我们引入了一种变分方法来生成这些贝叶斯伪标签，包括学习阈值以自动选择高质量的伪标签。在本文的其余部分中，我们展示了伪标记及其广义形式贝叶斯伪标记在医学图像的半监督分割中的应用。具体来说，我们关注：（1）从 CT 体积中对肺血管进行 3D 二元分割； (2) MRI 体积中脑肿瘤的 2D 多类分割； (3) 从 MRI 体积中对全脑肿瘤进行 3D 二元分割； (4) 根据 MRI 体积对前列腺进行 3D 二元分割。我们进一步证明伪标签可以增强学习表示的鲁棒性。该代码发布在以下 GitHub 存储库中： 。