Bayesian Hierarchical Models for Counterfactual Estimation (2301.08833v1)

Abstract: Counterfactual explanations utilize feature perturbations to analyze the outcome of an original decision and recommend an actionable recourse. We argue that it is beneficial to provide several alternative explanations rather than a single point solution and propose a probabilistic paradigm to estimate a diverse set of counterfactuals. Specifically, we treat the perturbations as random variables endowed with prior distribution functions. This allows sampling multiple counterfactuals from the posterior density, with the added benefit of incorporating inductive biases, preserving domain specific constraints and quantifying uncertainty in estimates. More importantly, we leverage Bayesian hierarchical modeling to share information across different subgroups of a population, which can both improve robustness and measure fairness. A gradient based sampler with superior convergence characteristics efficiently computes the posterior samples. Experiments across several datasets demonstrate that the counterfactuals estimated using our approach are valid, sparse, diverse and feasible.

• The paper introduces a probabilistic Bayesian framework that generates diverse, sparse, and valid counterfactual explanations.
• The methodology uses hierarchical modeling with Hamiltonian Monte Carlo to efficiently explore complex posterior distributions while respecting domain constraints.
• Experimental results on datasets like Adult Income and German Credit demonstrate enhanced robustness and fairness compared to traditional point-estimation methods.

Bayesian Hierarchical Models for Counterfactual Estimation: A Probabilistic Approach to Generating Diverse, Sparse, and Valid Counterfactuals

Introduction to Bayesian Counterfactuals

Counterfactual explanations have emerged as a key tool for interpreting decision-making models, especially in high-stakes domains where understanding the rationale behind predictions is crucial. Traditional methods often aim to find a single modification to input features that would result in a desired outcome, potentially overlooking the multifaceted nature of decision-making. This paper introduces a novel Bayesian framework for generating a diverse set of counterfactual explanations. By treating the perturbations as random variables with prior distributions, it leverages Bayesian hierarchical modeling to not only generate multiple plausible counterfactuals but also to handle uncertainty, incorporate inductive biases, and respect domain-specific constraints efficiently through a gradient-based sampler.

Methodology

The proposed approach diverges from the single-point estimation prevalent in previous works by adopting a distribution-oriented perspective. Here are the key components of the methodology:

• Probabilistic Paradigm: Counterfactual perturbations are modeled as random variables, allowing for the sampling of diverse counterfactual explanations from the posterior distribution. This is a significant shift from deterministic optimization models, enabling the capture of a wider range of possibilities.
• Bayesian Hierarchical Modeling: By employing a hierarchical structure, the model efficiently shares information across different subgroups of a population. This not only improves the robustness and validity of the counterfactuals generated but also facilitates the assessment of fairness across demographic subgroups.
• Sampling Mechanism: Due to the intractable nature of the posterior distribution, Hamiltonian Monte Carlo (HMC), specifically its No-U-Turn Sampler variant, is utilized for efficient computation of posterior samples. The use of gradient information in HMC allows for effective exploration of the parameter space even in complex models.

Experimental Evaluation

The efficacy of the proposed method is demonstrated through experiments on several datasets, including Adult Income, German Credit, and HELOC. The results are promising, showing that the generated counterfactuals are not only valid and sparse but also exhibit diversity and feasibility. Quantitative comparison with traditional point-estimate methods further affirms the advantages of the probabilistic approach, particularly in generating diverse alternatives.

Implications and Future Directions

This research opens up several avenues for advancing the field of counterfactual explanations and interpretable AI at large. The ability to quantify uncertainty and incorporate prior knowledge directly addresses the limitations of current methods, potentially leading to more reliable and personalized decision-making tools. Furthermore, the hierarchical nature of the model lays the groundwork for more nuanced analyses of fairness and bias in AI systems.

Looking ahead, the framework could be expanded to include complex causal relationships and to handle non-differentiable classifiers. Such advancements would further broaden the applicability and effectiveness of Bayesian counterfactual methods in real-world scenarios.

Conclusion

The paper introduces a significant shift in counterfactual estimation by adopting a probabilistic, hierarchical Bayesian framework. This approach not only enhances the diversity and robustness of counterfactual explanations but also introduces flexibility in handling uncertainties and domain-specific constraints. Through comprehensive experimentation, the method's effectiveness in generating valid, sparse, and diverse counterfactuals is well demonstrated, marking a promising direction for future research in interpretable AI.