Mathematical theories can make AI explanations more explicit and testable, but no single theory makes every complex system explainable. Causal inference, cooperative game theory, statistical learning and information theory each help answer different questions about a model. Their results can clarify how a model produced a prediction; they do not, by themselves, prove what caused the real-world outcome represented by that prediction.
What does it mean to explain an AI system?
“Explainable AI” is not one settled mathematical objective. A person might want to know why a model made one prediction, which inputs generally matter to it, whether its behavior changes under an intervention, or whether its conclusions reflect a real-world mechanism. Those are different questions, and an explanation suited to one may not answer another.
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It also matters who is asking. A developer investigating a failure may need a detailed account of model behavior; a clinician or other decision-maker may need a concise explanation relevant to a particular case. David S. Watson and Luciano Floridi’s 2020/2021 paper, The explanation game: a formal framework for interpretable machine learning, treats explanation as a collaborative process of questions and answers. It considers trade-offs among accuracy, simplicity and relevance, with the appropriate level of detail depending on the questioner’s needs.
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That framework points toward a conditionally optimal set of explanations for a given context—not one universally best explanation. An exact account can be too complicated to help, while a simple account can leave out details needed for accuracy. For an image classifier, for example, a person may learn more from a description of relevant edges or shapes than from pixel-level details.
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Which mathematical approaches answer which questions?
The methods differ in what they explain and what their results mean. The table describes broad uses, not a ranking: the reviewed sources do not establish a universal benchmark for comparing all approaches.
| Approach | Question it can help address | What the result describes | Important limit |
|---|---|---|---|
| Statistical learning and local approximation, including LIME | How does a model behave near this particular prediction? | A locally weighted linear regression used to approximate the model’s behavior around that case, as described in the 2025 EMS Magazine article on explainable AI and mathematics. | A local approximation does not necessarily describe the whole model or establish why an input caused a real-world outcome. |
| Cooperative game theory, including SHAP | How can a prediction be divided into feature contributions under a specified explanation setup? | Shapley values allocate contributions among features in relation to a model prediction. | The allocation depends on the model and explanation setup. A contribution is not, by itself, proof of real-world causation. |
| Causal inference and interventionism | What might change in a model’s output if an input or condition were changed? | An intervention-based account of relationships under specified assumptions; causal ideas also inform Watson and Floridi’s explanation-game framework. | Changing an input in a model is not automatically equivalent to intervening on the real system that generated the data. |
| Information theory | How much information do variables or an explanation share with predictions? | Quantities such as mutual information can describe statistical information shared between variables. | Information shared with a prediction does not on its own say whether the relationship is causal, useful to a particular audience or scientifically correct. |
| Information geometry | How can the structure of statistical models and their representations be analyzed? | A mathematical lens identified among directions relevant to explainable AI. | It is not a turnkey explanation; the method has to fit the question and available data. |
The EMS Magazine article, “Explainable artificial intelligence and mathematics: What lies behind? Let us focus” (2025), discusses LIME, SHAP, causal inference, information theory and information geometry as mathematical directions for XAI. They are complementary tools, not interchangeable answers.
Does an explanation show why a model made a prediction?
It can show something about the model, but the word “why” needs care. A feature-attribution method may describe how inputs contribute to a prediction under a particular setup. That is a model-level explanation: evidence about the model’s behavior. It is not automatically a causal account of the outside world.
For example, if an explanation assigns importance to an input feature, that does not establish that changing the corresponding real-world factor would change the outcome. The feature may be associated with the prediction in the data, or the attribution may reflect how the explanation method represents the model. The causal claim requires evidence beyond the attribution itself.
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This distinction is central to the August 6, 2026 Nature Communications perspective “Explainable AI: learning from the learners,” by Ricardo Vinuesa, Steven L. Brunton and Gianmarco Mengaldo. It separates explanations of model inputs, representations or internal components from system-level claims about the physical process or world being modeled. The authors characterize explainability analyses as hypothesis-generating unless additional validation supports the proposed mechanism.
What makes an explanation useful—and what can go wrong?
A useful explanation needs to fit both the model and the reader’s question. Accuracy, simplicity and relevance can pull in different directions: a highly detailed account may be faithful but hard to use, while an accessible summary may omit important qualifications. The explanation-game framework makes these trade-offs explicit rather than assuming that one output is best for every audience.
- Accuracy or fidelity: Does the explanation faithfully represent the model or prediction it describes?
- Simplicity: Can the intended reader understand it without stripping away details essential to its meaning?
- Relevance: Does it address the reader’s actual question, rather than merely listing available model details?
- Abstraction: Is the explanation at a useful level, such as shapes in an image rather than individual pixels?
- Scope: Does it describe one prediction, a local region of behavior or the model more broadly?
- Evidence: Is a proposed claim about the real system independently tested, rather than inferred from model attribution alone?
There is another distinction: some models are designed to be interpretable, while post-hoc methods attempt to explain a model after it has been trained. A transparent model’s structure can make its operation easier to inspect, but it still needs to be evaluated for whether it represents the task adequately. A post-hoc explanation can help analyze a more complex model, but its account is an additional approximation or analysis—not a guarantee that the model is understandable or correct.
The quality of any explanation is also limited by the model and data behind it. A model that is inaccurate or trained on unrepresentative data cannot be made reliable simply by adding an explanation. The 2026 Nature Communications perspective states, “The utility of any explainability analysis is bounded by the quality of the underlying model.”
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How can a model explanation support a scientific claim?
To move from a model-level explanation to a claim about the world, researchers need independent evidence for the proposed mechanism. The 2026 Nature Communications perspective identifies targeted simulations, physical experiments, representative data, robustness checks, and consistency with known constraints or governing models as relevant forms of validation.
- State the claim precisely. Separate “the model relied on this feature” from “this real-world factor caused the outcome.”
- Check the model’s foundation. Assess whether its data are representative and whether the model performs adequately for the cases at issue.
- Test stability. Use robustness checks to see whether the explanation changes under relevant variations in data or analysis.
- Test the proposed mechanism independently. Where appropriate, use targeted simulations or physical experiments rather than treating an attribution as confirmation.
- Compare with established constraints. Check whether the interpretation is consistent with known constraints or governing models, and investigate discrepancies.
These checks do not turn every explanation into a complete causal account; they provide additional evidence against which a proposed account can be judged. NIST’s David A. Broniatowski discusses the psychological foundations of explainability and interpretability in AI in NISTIR 8367, published April 12, 2021—a reminder that an explanation is also meant to be interpreted by people, not merely computed from a model.
So, could mathematics be the key?
Mathematics is part of the answer because it lets researchers define explanation goals, analyze feature contributions, formalize interventions and describe information shared between variables. Its most important contribution may be making the limits of an explanation clearer: what question it answers, what assumptions it relies on and whether it describes a model or supports a claim about the world.
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1Scan for outdated or missing drivers - takes under a minute2Repair Windows errors before they cause bigger problems3Fix the driver behind crashes, sound loss and screen glitchesThere is no single mathematical method that settles explainability for every model and audience. The right approach depends on the question, model, data and required level of evidence. For scientific or consequential claims, model explanations are a starting point for investigation; they need independent validation before being treated as explanations of real-world mechanisms.
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