Explanatory relationships between data and hypotheses have been suggested to play a role in the formation of posterior probabilities. This suggestion was tested in a toy environment and supported by simulations by David H. Glass. We here put forward a variety of inference to the best explanation approaches for determining posterior probabilities by intertwining Bayesian and inference to the best explanation approaches. We then simulate their performances for the estimation of parameters in the Brock and Hommes agent-based model for asset pricing in finance. We find that performances depend on circumstances and also on the evaluation metric. However, most of the time our suggested approaches outperform the Bayesian approach. • Bayesian posteriors and Inference to the Best Explanation are intertwined. • Inference to the Best Explanation posteriors outperform the Bayesian posterior. • Parameters are estimated for the Brock and Hommes model for asset pricing. • Comparison of explanation-based and probabilistic inference.

Deep neural networks (DNNs) defy the classical bias-variance trade-off: adding parameters to a DNN that interpolates its training data will typically improve its generalization performance. Explaining the mechanism behind this ``benign overfitting'' in deep networks remains an outstanding challenge. Here, we study the last hidden layer representations of various state-of-the-art convolutional neural networks and find that if the last hidden representation is wide enough, its neurons tend to split into groups that carry identical information and differ from each other only by statistically independent noise. The number of such groups increases linearly with the width of the layer, but only if the width is above a critical value. We show that redundant neurons appear only when the training is regularized and the training error is zero.

Deep convolutional networks (DCNs) learn meaningful representations where data that share the same abstract characteristics are positioned closer and closer. Understanding these representations and how they are generated is of unquestioned practical and theoretical interest. In this work we study the evolution of the probability density of the ImageNet dataset across the hidden layers in some state-of-the-art DCNs. We find that the initial layers generate a unimodal probability density getting rid of any structure irrelevant for classification. In subsequent layers density peaks arise in a hierarchical fashion that mirrors the semantic hierarchy of the concepts. Density peaks corresponding to single categories appear only close to the output and via a very sharp transition which resembles the nucleation process of a heterogeneous liquid. This process leaves a footprint in the probability density of the output layer where the topography of the peaks allows reconstructing the semantic relationships of the categories.

We introduce a principled and theoretically sound spectral method for k-way clustering in signed graphs, where the affinity measure between nodes takes either positive or negative values. Our approach is motivated by social balance theory, where the task of clustering aims to decompose the network into disjoint groups such that individuals within the same group are connected by as many positive edges as possible, while individuals from different groups are connected by as many negative edges as possible. Our algorithm relies on a generalized eigenproblem formulation inspired by recent work on constrained clustering. We provide theoretical guarantees for our approach in the setting of a signed stochastic block model, by leveraging tools from matrix perturbation theory and random matrix theory. An extensive set of numerical experiments on both synthetic and real data shows that our approach compares favorably with state-of-the-art methods for signed clustering, especially for large number of clusters and sparse measurement graphs.