Conformal Prediction

Unlike Bayesian methods, conformal prediction makes no assumptions about the underlying data distribution. The coverage guarantee holds for any data-generating process, even with heteroscedastic noise or heavy-tailed distributions. This is achieved through exchangeability rather than parametric assumptions.

• Works with any black-box model
• No need to model error distributions
• Robust to distributional shifts in the calibration set

For a user-specified significance level α (e.g., 0.1), conformal prediction guarantees that the true label will fall within the prediction set with probability at least 1 − α over the randomness of both the calibration and test data.

• Set α = 0.1 → 90% coverage probability
• Guarantee is marginal (averaged over all test points)
• Finite-sample validity: holds for any calibration set size n

The engine of conformal prediction is a nonconformity score function that quantifies how unusual a candidate label is given the training data. Common choices include:

• Regression: Absolute residual |y − ŷ|
• Classification: 1 − softmax(y_true) or logit margin
• Custom: Any function A(z, D) measuring strangeness

The flexibility of this score function allows domain experts to encode task-specific notions of uncertainty.

The most practical variant, inductive (split) conformal prediction, avoids retraining the model by holding out a dedicated calibration set. The procedure:

1. Train model on proper training set
2. Compute nonconformity scores on held-out calibration set
3. For a new test point, compute scores for all candidate labels
4. Include labels whose score is below the (1−α) quantile of calibration scores

This makes conformal prediction computationally tractable for deep learning models.

Conformal prediction naturally produces instance-adaptive prediction intervals. When the model is uncertain (high nonconformity), the prediction set expands; when confident, it shrinks. This contrasts with fixed-width confidence intervals.

• Easy examples → smaller sets (often singletons)
• Ambiguous or out-of-distribution examples → larger sets or empty sets
• Empty set signals that no label is plausible, flagging distributional shift

Standard conformal prediction provides only marginal coverage, which can fail for important subgroups. Advanced variants address this:

• Mondrian conformal prediction: Guarantees coverage conditional on a discrete category (e.g., per-class coverage in classification)
• Conformalized quantile regression (CQR): Produces adaptive intervals that approximate conditional coverage for continuous covariates
• Weighted conformal prediction: Handles covariate shift by reweighting calibration scores

The process of aligning a model's predicted probability with its actual empirical accuracy. While conformal prediction provides a distribution-free coverage guarantee, calibration ensures that a 90% confidence score genuinely reflects a 90% chance of correctness. Expected Calibration Error (ECE) is the standard metric for measuring miscalibration. Unlike conformal prediction, calibration methods like Platt scaling or temperature scaling assume the model's scores are meaningful probabilities, which is not always true for modern neural networks.

A metric quantifying the uncertainty in a language model's output distribution, used as a predictive signal for detecting confabulated text. Conformal prediction can wrap these entropy scores to produce statistically valid prediction sets of possible hallucinations. Key techniques include:

• Semantic entropy: Clusters token-level predictions by meaning before calculating uncertainty
• Predictive entropy: Measures raw token probability dispersion
• Length-normalized entropy: Accounts for varying sequence lengths

A statistical technique that estimates conditional quantiles of a response variable, directly related to conformal prediction's conformalized quantile regression (CQR) variant. While standard conformal prediction produces fixed-width intervals, CQR adapts interval widths based on input features, yielding tighter prediction bands in low-heteroscedasticity regions and wider bands where uncertainty is high. This is particularly valuable for time-series forecasting and heteroscedastic data.

A class of statistical methods that make no assumptions about the underlying data distribution. Conformal prediction is a leading example, requiring only exchangeability of the calibration and test data. This contrasts with:

• Parametric methods: Assume Gaussian or other specific distributions
• Bayesian methods: Require prior distribution specification
• Bootstrap methods: Rely on asymptotic approximations The trade-off is that distribution-free guarantees are often conservative, producing wider intervals than parametric alternatives when assumptions hold.

A computationally efficient variant that splits the training data into a proper training set and a calibration set. The model is trained once on the proper training set, and nonconformity scores are computed on the calibration set. This avoids the retraining overhead of full conformal prediction while maintaining the same coverage guarantee. ICP is the practical standard for deep learning applications where retraining is prohibitively expensive.

A real-valued function that quantifies how unusual a new example is relative to a calibration set. Common choices include:

• Absolute residual: For regression, the absolute difference between prediction and true value
• 1 - softmax score: For classification, the complement of the predicted class probability
• Mahalanobis distance: For detecting out-of-distribution inputs The choice of nonconformity measure determines the efficiency of the resulting prediction sets—how small the sets are while maintaining coverage.