Counterfactual Validity

Validity is a strict boolean metric: a counterfactual instance is either valid or invalid. There is no partial credit. A generated instance is considered valid if and only if the model's predicted class for that instance matches the target outcome. This makes it the most fundamental evaluation metric, as an invalid counterfactual fails its primary purpose of providing recourse. All other quality metrics—proximity, sparsity, plausibility—are secondary and only meaningful once validity is established.

Formally, for a model f and a target class y', a counterfactual instance x_cf is valid if:

f(x_cf) = y'

This is often implemented as a validity rate across a test set:

Validity = (1/N) * Σ I[f(x_cf_i) = y'_i]

where I is the indicator function and N is the number of test instances. A perfect validity rate of 1.0 means the generation algorithm always produces instances that flip the prediction.

Validity is always evaluated relative to a predefined target outcome. This target can be:

• Binary flip: Changing a loan denial to an approval
• Multi-class shift: Reclassifying a medical image from 'benign' to a specific 'malignant' subtype
• Regression target: Achieving a credit score above a threshold (e.g., > 700)

For regression models, validity is defined as the predicted value crossing a specified decision threshold, not just moving in the desired direction.

A valid counterfactual must cross the model's decision boundary into the target class region. This geometric interpretation reveals why validity can fail:

• Insufficient perturbation: The feature changes were too small to reach the boundary
• Wrong direction: Features were modified in a direction that doesn't lead to the target class
• Disconnected regions: The target class region may be unreachable from the query point without passing through other classes

Understanding decision boundary topology is critical for diagnosing systematic validity failures in generation algorithms.

There is an inherent tension between validity and proximity. The closest counterfactual to the original instance may lie exactly on the decision boundary, where small numerical perturbations can cause the prediction to flip back. This leads to:

• Low robustness: The counterfactual becomes invalid after minor model updates
• Adversarial instances: Valid counterfactuals that are not realistic or actionable

High-quality generation algorithms often sacrifice minimal proximity to find counterfactuals deeper inside the target class region, improving both validity stability and plausibility.

In deployed recourse systems, validity must be verified at multiple stages:

• Generation time: Does the algorithm output flip the prediction?
• Post-hoc verification: Re-query the model with the generated instance to confirm
• Model update monitoring: Track validity drift after model retraining to measure recourse robustness

A counterfactual that was valid at generation time but becomes invalid after a model update represents broken recourse, which can have regulatory implications in lending and hiring applications.

The process of translating a valid counterfactual into a set of actionable recommendations for the end-user. While validity confirms the prediction flipped, recourse ensures the user can actually implement the suggested changes.

• Bridges the gap between explanation and action
• Requires validity as a prerequisite; an invalid flip offers no recourse
• Often constrained by an action set of permissible modifications

A metric quantifying the distance between the original instance and the valid counterfactual, typically using L1 (Manhattan) or L2 (Euclidean) norms. A lower proximity score indicates a smaller, more interpretable change.

• Minimizing proximity is a core objective in counterfactual search
• Trade-off exists between validity and proximity
• High proximity may signal an adversarial example rather than a true explanation

A valid counterfactual that lies within the high-density region of the training data distribution. Plausibility ensures the explanation is realistic and not an artifact of the model's blind spots.

• Measured using Mahalanobis distance or density estimation
• Prevents generating nonsensical feature combinations
• Critical for maintaining user trust in the explanation system

The property that a counterfactual remains valid after model retraining or minor updates. A fragile counterfactual that fails post-update undermines the reliability of the explanation system.

• Evaluates the stability of the decision boundary
• Robust recourse requires generating counterfactuals with a margin beyond the boundary
• Key concern for models in continuous learning environments

Generating a set of multiple, distinct valid counterfactuals instead of a single point. Diversity provides users with alternative paths to a desired outcome, avoiding a single recommendation that may be infeasible.

• Uses determinantal point processes or latent space sampling
• Balances validity with intra-set feature dissimilarity
• Empowers user choice in the recourse process

A hard rule encoded into the generation algorithm that prevents modification of immutable features (e.g., age, birthplace) or enforces causal monotonicity (e.g., education level cannot decrease).

• Defined by a formal action set
• Ensures validity is achieved through actionable changes only
• Violating constraints produces a valid but useless counterfactual