Title: Revisiting Information-Based Clustering with Pseudo-Posterior Models. Abstract: Maximization of mutual information (MI) between the network's input and output motivates standard losses for unsupervised discriminative clustering enforcing "decisiveness" and "fairness". In the context of common softmax models, we clarify several general properties of such discriminative losses that were previously not well understood: the relation to K-means, or lack thereof, and "margin-maximization". In particular, we show that "desiciveness" without the extra regularization term can lead to poor classification margins. Also, non-convexity of information-based losses motivates us to focus on self-supervised approaches introducing effective higher-order optimization algorithms with auxiliary variables. Addressing limitations of existing formulations, we propose a new self-supervised loss with soft auxiliary variables, or "pseudo-confidence" estimates. In particular, we introduce "strong" fairness and motivate the "reverse" cross-entropy as a robust loss for network training from noisy pseudo-confidence estimates. The latter is efficiently computed using variational inference - we derive a new EM algorithm with closed-form solutions for E and M steps. Empirically, our algorithm improves the performance of earlier methods for information-based clustering.