Post-hoc Stochastic Concept Bottleneck Models

TL;DR

This paper introduces Post-hoc Stochastic Concept Bottleneck Models (PSCBMs), a lightweight method that augments any pre-trained CBM with a multivariate normal distribution over concepts by adding only a small covariance-prediction module, without retraining the backbone model.

Abstract

Concept Bottleneck Models (CBMs) are interpretable models that predict the target variable through high-level human-understandable concepts, allowing users to intervene on mispredicted concepts to adjust the final output. While recent work has shown that modeling dependencies between concepts can improve CBM performance, especially under interventions, such approaches typically require retraining the entire model, which may be infeasible when access to the original data or compute is limited. In this paper, we introduce Post-hoc Stochastic Concept Bottleneck Models (PSCBMs), a lightweight method that augments any pre-trained CBM with a multivariate normal distribution over concepts by adding only a small covariance-prediction module, without retraining the backbone model. We propose two training strategies and show on real-world data that PSCBMs consistently match or improve both concept and target accuracy over standard CBMs at test time. Furthermore, we show that due to the modeling of concept dependencies, PSCBMs perform much better than CBMs under interventions, while remaining far more efficient than retraining a similar stochastic model from scratch.

Figures (4)

• Figure 1: Comparison of CBM (left), SCBM (right), and our proposed PSCBM (right, red block). All methods input ${\bm{x}}$ to an encoder to produce the feature vector ${\bm{z}}$ from which concepts are obtained and fed to the target predictor. CBM directly predicts concept values $\hat{{\bm{c}}}$, while in SCBM, the predictor outputs an expected value $\boldsymbol{\mu}$ and a covariance matrix $\boldsymbol{\Sigma}$ that define a multivariate normal distribution to sample from. PSCBM incorporates the $\boldsymbol{\Sigma}$ predictor (red box) to a pre-trained CBM.
• Figure 2: Intervention curves of Concept and Target Accuracy for PSCBM models and baselines when concept uncertainty policy is used. Confidence intervals are thinner than the lines.
• Figure 3: Concept and Target Accuracy for amortized PSCBM models and baselines when Concept Uncertainty Policy is used. The shadings represent the standard deviation.
• Figure 4: Concept and Target Accuracy for PSCBM models and baselines when Random policy is used. The shadings represent the standard deviation.