Estimating the confidence of deep neural network predictions is crucial for safe deployment in high-stakes applications. While softmax probabilities are commonly used, they are often poorly calibrated, and existing calibration methods have been shown to be detrimental to failure prediction. In this paper, we propose using information-theoretic measures to estimate prediction confidence in a post-hoc manner, without modifying network architecture or training. Specifically, we compare three pointwise information (PI) measures: pointwise mutual information (PMI), pointwise $\mathcal{V}$-information (PVI), and the recently proposed pointwise sliced mutual information (PSI). These measures are theoretically grounded in their relevance to predictive uncertainty, with properties such as invariance, convergence rates, and sensitivity to geometric attributes like margin and intrinsic dimensionality. Through extensive experiments on benchmark computer vision models and datasets, we find that PVI consistently outperforms PMI, PSI and existing post-hoc baselines in failure prediction across metrics. For confidence calibration, PVI matches the performance of temperature-scaled softmax, which is already regarded as a highly effective baseline. This indicates that PVI achieves superior failure prediction without compromising its calibration performance. This aligns with our theoretical insights, which suggest that PVI offers the most balanced trade-offs.

In this paper, we study the memorization and generalization behaviour of deep neural networks (DNNs) using sliced mutual information (SMI), which is the average of the mutual information (MI) between one-dimensional random projections. We argue that the SMI between features in a DNN ($T$) and ground truth labels ($Y$), $SI(T;Y)$, can be seen as a form of usable information that the features contain about the labels. We show theoretically that $SI(T;Y)$ can encode geometric properties of the feature distribution, such as its spherical soft-margin and intrinsic dimensionality, in a way that MI cannot. Additionally, we present empirical evidence showing how $SI(T;Y)$ can capture memorization and generalization in DNNs. In particular, we find that, in the presence of label noise, all layers start to memorize but the earlier layers stabilize more quickly than the deeper layers. Finally, we point out that, in the context of Bayesian Neural Networks, the SMI between the penultimate layer and the output represents the worst case uncertainty of the network’s output.

When deploying deep learning models such as convolutional neural networks (CNNs) in safety-critical domains, it is important to understand the predictions made by these black-box models. While many different approaches have been taken to improve the interpretability of deep models, most methods lack theoretical properties, making it hard to justify the correctness of the explanations provided. In this paper, we take an information-theoretic approach to understand why CNNs make their predictions. Building upon sliced mutual information, we propose pointwise sliced mutual information (PSI) as a tool for measuring the amount of useful information that a feature has about the label for a single instance. We theoretically justify the use of PSI for explaining predictions made by CNNs through the connection with margin. We show that PSI works as an explainability tool in two ways: (i) Fiber-wise PSI constructs a saliency map that highlights regions in the image which are important in predicting the labels; (ii) Sample-wise PSI provides confidence scores for predictions on the labels.

Along with the practical success of deep neural networks, several theories have been proposed to explain their excellent generalization behaviour. One such theory is information bottleneck, using mutual information (MI) as a measure to understand the learning dynamics of these black-box models. However, estimating MI in high dimensions and in deterministic settings is problematic, often resulting in widely varying estimates for different estimators. This paper takes an alternative approach to analyze the behaviour of deep models by using a recently proposed information measure known as sliced mutual information (SMI), which is more computationally efficient to estimate than MI in high dimensions. We theoretically connect SMI to the classifier margin, thereby showcasing its ability to encode geometric properties of the feature distribution. We also study SMI empirically and demonstrate that the SMI between the hidden layer and the labels encodes information about the network’s ability to predict labels correctly.