Minimal Sufficient Explanations

The core tenet of minimal sufficient explanations is that explanatory power should not be diluted by irrelevant information. A minimal explanation strips away all features, reasoning steps, or evidence that do not causally contribute to the prediction.

• Sparsity constraint: The explanation must use the fewest possible features or tokens
• Sufficiency guarantee: Despite its brevity, the explanation must logically entail the prediction
• Contrast with completeness: A complete explanation may include all contributing factors; a minimal one includes only the decisive ones

This principle is directly inspired by Occam's Razor and formalized in works on anchor explanations and prime implicant explanations.

A minimal sufficient explanation must satisfy two logical conditions simultaneously. Sufficiency means the cited evidence alone guarantees the prediction; necessity means removing any cited element would break that guarantee.

• Sufficient condition: If features A and B are present, the model always predicts class X
• Necessary condition: If feature A is removed, the prediction changes, even if B remains
• Minimality: The set contains no redundant elements—every cited feature is necessary for sufficiency

This dual requirement prevents explanations that are either too vague (insufficient) or bloated with non-causal correlations (not minimal).

Derived from Boolean circuit logic, prime implicant explanations identify the minimal set of input features that logically force a specific model output. A prime implicant is a conjunction of literals that implies the output and cannot be further reduced.

• Formal logic grounding: Rooted in Quine-McCluskey algorithm for circuit minimization
• Deterministic guarantee: Unlike statistical anchors, prime implicants provide absolute logical sufficiency
• Applicability: Best suited for models with discretizable or binarizable input spaces, including decision trees and binarized neural networks

This approach provides the strongest possible minimality guarantee—no smaller set of features can logically determine the output.

Minimal sufficient explanations gain practical utility when framed contrastively—explaining why outcome A was predicted instead of outcome B. This narrows the explanatory burden to only the differentiating factors.

• Foil specification: The user or system specifies a contrast class (e.g., 'Why denied instead of approved?')
• Minimal contrast set: Only features that differ between the actual and contrast prediction are cited
• Cognitive alignment: Humans naturally seek contrastive explanations in decision-making contexts

This approach dramatically reduces explanation length by ignoring features that are equally present in both outcomes.

Quantifying whether an explanation is truly minimal requires specialized metrics beyond standard faithfulness measures.

• Compression ratio: The size of the explanation relative to the full input feature space
• Sufficiency precision: The percentage of perturbations where the cited features alone reproduce the original prediction
• Minimality violation rate: How often a subset of the explanation's features also proves sufficient
• Human simulatability: Can a human reproduce the model's decision using only the minimal explanation?

These metrics ensure that 'minimal' is not merely a qualitative claim but a measurable property of the explanation.

A faithful rationale accurately mirrors the true internal reasoning process of the model, not just a plausible-sounding story. This is the gold standard for minimal sufficient explanations, as a short justification is useless if it describes a logic path the model didn't actually use.

• Key Distinction: Faithfulness vs. Plausibility
• Challenge: Complex neural networks often use distributed representations that don't map cleanly to human-interpretable logic.
• Evaluation: Requires techniques like input erasure or counterfactual simulation to verify the rationale's causal influence on the prediction.

Contrastive explanations answer the question: 'Why did the model predict outcome A instead of outcome B?' This format is inherently minimal, as it focuses only on the diagnostic features that differentiate the two outcomes.

• Structure: 'The loan was denied because your debt-to-income ratio was 45%, whereas it would have been approved if it were below 43%.'
• Cognitive Alignment: Humans naturally seek contrastive 'why' answers, making this format highly intuitive.
• Relation to MSE: A contrastive frame forces the system to identify the smallest set of features that flip the decision.

Anchors are high-precision if-then rules that sufficiently 'anchor' a prediction locally. The rule states that if a specific set of conditions is met, the prediction will remain fixed regardless of changes to other feature values.

• Example: 'If the applicant's credit score > 720 and loan amount < $300k, the approval decision is guaranteed.'
• Sufficiency: Anchors provide a formal guarantee of sufficiency within a defined perturbation neighborhood.
• Minimality: The algorithm searches for the shortest anchor rule with the highest coverage, directly optimizing for minimal sufficient conditions.

The regulatory requirement under the General Data Protection Regulation for providing 'meaningful information about the logic involved' in automated decisions. This legal mandate is a primary driver for minimal sufficient explanations in enterprise systems.

• Recital 71: Grants the right 'to obtain an explanation of the decision reached.'
• Practical Tension: The law demands both completeness (meaningful) and accessibility (understandable to the data subject).
• MSE Alignment: A minimal sufficient explanation balances these competing demands by providing the shortest justification that is still legally defensible.

The ability of a human observer to use a model's explanation to correctly anticipate the model's output on a new, unseen input. This is a rigorous behavioral test for the sufficiency of an explanation.

• Test Protocol: A user reads the explanation, then predicts the model's output for a held-out example.
• Forward Simulation: If the explanation is truly sufficient, the human should be able to run the described logic forward and arrive at the correct prediction.
• MSE Metric: A minimal explanation that fails the simulatability test is insufficient; one that passes is both minimal and complete.

Rationales that not only explain a decision but also provide the user with clear steps to change the outcome in the future. This extends minimal sufficient explanations from passive understanding to active recourse.

• Recourse Focus: 'Your application was denied. To qualify, increase your savings balance by $5,000.'
• Causal Constraint: The suggested actions must be causally valid and within the user's power to execute.
• Minimality in Action: The best actionable explanation identifies the single smallest change that flips the decision, embodying the principle of minimal intervention.