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Explainable Bayesian deep learning through input-skip Latent Binary Bayesian Neural Networks

Eirik Høyheim, Lars Skaaret-Lund, Solve Sæbø, Aliaksandr Hubin

TL;DR

This work introduces input-skip latent binary Bayesian neural networks (ISLaB) to address interpretability and uncertainty in deep models by allowing covariates to skip to any layer and by learning sparse, structure-aware networks through latent inclusion indicators. The approach combines spike-and-slab priors with variational inference to yield highly compact models that retain predictive accuracy while enabling global and local explanations via active paths. ISLaB demonstrates exceptional sparsity across diverse datasets, achieving near-linear behavior on simple problems and capturing essential nonlinear interactions when needed, with demonstrated improvements in calibration and interpretability. The methodology offers a practical framework for uncertainty-aware explanations in Bayesian neural networks, enabling reliable covariate-level insights without relying on post hoc tools, and shows significant potential for efficient deployment in real-world settings.

Abstract

Modeling natural phenomena with artificial neural networks (ANNs) often provides highly accurate predictions. However, ANNs often suffer from over-parameterization, complicating interpretation and raising uncertainty issues. Bayesian neural networks (BNNs) address the latter by representing weights as probability distributions, allowing for predictive uncertainty evaluation. Latent binary Bayesian neural networks (LBBNNs) further handle structural uncertainty and sparsify models by removing redundant weights. This article advances LBBNNs by enabling covariates to skip to any succeeding layer or be excluded, simplifying networks and clarifying input impacts on predictions. Ultimately, a linear model or even a constant can be found to be optimal for a specific problem at hand. Furthermore, the input-skip LBBNN approach reduces network density significantly compared to standard LBBNNs, achieving over 99% reduction for small networks and over 99.9% for larger ones, while still maintaining high predictive accuracy and uncertainty measurement. For example, on MNIST, we reached 97% accuracy and great calibration with just 935 weights, reaching state-of-the-art for compression of neural networks. Furthermore, the proposed method accurately identifies the true covariates and adjusts for system non-linearity. The main contribution is the introduction of active paths, enhancing directly designed global and local explanations within the LBBNN framework, that have theoretical guarantees and do not require post hoc external tools for explanations.

Explainable Bayesian deep learning through input-skip Latent Binary Bayesian Neural Networks

TL;DR

This work introduces input-skip latent binary Bayesian neural networks (ISLaB) to address interpretability and uncertainty in deep models by allowing covariates to skip to any layer and by learning sparse, structure-aware networks through latent inclusion indicators. The approach combines spike-and-slab priors with variational inference to yield highly compact models that retain predictive accuracy while enabling global and local explanations via active paths. ISLaB demonstrates exceptional sparsity across diverse datasets, achieving near-linear behavior on simple problems and capturing essential nonlinear interactions when needed, with demonstrated improvements in calibration and interpretability. The methodology offers a practical framework for uncertainty-aware explanations in Bayesian neural networks, enabling reliable covariate-level insights without relying on post hoc tools, and shows significant potential for efficient deployment in real-world settings.

Abstract

Modeling natural phenomena with artificial neural networks (ANNs) often provides highly accurate predictions. However, ANNs often suffer from over-parameterization, complicating interpretation and raising uncertainty issues. Bayesian neural networks (BNNs) address the latter by representing weights as probability distributions, allowing for predictive uncertainty evaluation. Latent binary Bayesian neural networks (LBBNNs) further handle structural uncertainty and sparsify models by removing redundant weights. This article advances LBBNNs by enabling covariates to skip to any succeeding layer or be excluded, simplifying networks and clarifying input impacts on predictions. Ultimately, a linear model or even a constant can be found to be optimal for a specific problem at hand. Furthermore, the input-skip LBBNN approach reduces network density significantly compared to standard LBBNNs, achieving over 99% reduction for small networks and over 99.9% for larger ones, while still maintaining high predictive accuracy and uncertainty measurement. For example, on MNIST, we reached 97% accuracy and great calibration with just 935 weights, reaching state-of-the-art for compression of neural networks. Furthermore, the proposed method accurately identifies the true covariates and adjusts for system non-linearity. The main contribution is the introduction of active paths, enhancing directly designed global and local explanations within the LBBNN framework, that have theoretical guarantees and do not require post hoc external tools for explanations.

Paper Structure

This paper contains 29 sections, 3 theorems, 27 equations, 25 figures, 19 tables.

Table of Contents

  1. Introduction
  2. Materials and Methods
  3. Bayesian neural networks
  4. Latent Binary Bayesian Neural Networks
  5. Input-skip in Latent Binary Bayesian Neural Networks
  6. Active Paths
  7. Max and Average Depth in ISLaB Models
  8. Global Understanding of Predictions
  9. Local Explainability
  10. Local Explainability through gradients
  11. Experiments
  12. Experimental set-up
  13. Global explanations and model performance
  14. Simulated linear data
  15. Simulated non-linear data
  16. ...and 14 more sections

Key Result

Proposition 1

Let the linear predictor parameter $\zeta(\textbf{x}_i)$ of the probability distribution of the responses be modeled by a neural network model with a fixed architecture and ReLU activations, parameterized by known weights $\textbf{W}$. Assume with no loss of generality ${\textit{y}}_i \in \mathbb{R} where $\beta_j \in \mathbb{R}$ are effective, explainable slope coefficients derived from $\textbf{

Figures (25)

  • Figure 1: A simple ANN architecture without bias nodes.
  • Figure 2: Illustration of simple ANN (on the left) and BNN (on the right) architectures.
  • Figure 3: Illustration of the difference between BNNs (left) and LBBNNs (right).
  • Figure 4: Concatenation is used to allow the input to skip directly to any given hidden layer.
  • Figure 5: Example of nodes that are indirectly inactive.
  • ...and 20 more figures

Theorems & Definitions (10)

  • Proposition 1
  • proof
  • Remark 1
  • Remark 2
  • Remark 3
  • Corollary 1
  • proof
  • Remark 4
  • Corollary 2
  • proof