Conformal Predictive Distributions

Standard conformal prediction outputs a prediction set—an interval or collection of labels. A CPD upgrades this to a full predictive distribution function over the label space. Instead of asking 'Is this value in the set?', you can query 'What is the probability that the label is below this threshold?' This provides a richer, more granular uncertainty representation for downstream decision-making.

The core theoretical property is that the true cumulative distribution function (CDF) of the test label is stochastically dominated by the conformal predictive distribution. This means the CPD is conservatively valid: for any threshold, the probability that the true label falls below it is at least the CPD's estimated probability. This guarantee holds under the standard exchangeability assumption.

A CPD is built by evaluating a nonconformity measure for every possible label value. For regression, this often involves:

• Fitting a base predictor (e.g., mean regression) on proper training data
• Computing residuals on a calibration set
• For a new test point, the CPD at value y is the normalized rank of the nonconformity score of (x_test, y) among the calibration scores This process converts raw residuals into a valid probability distribution.

Conformalized Quantile Regression (CQR) is a specific, practical method for constructing a CPD. A base quantile regressor estimates conditional quantiles, which are then calibrated using conformal inference. The result is a predictive distribution that inherits the adaptive width of quantile regression—wider in high-noise regions, tighter in low-noise regions—while correcting any coverage deficiencies to achieve finite-sample validity.

To achieve exact distributional validity, CPDs often incorporate a randomization component that smooths the discrete jumps in the empirical CDF. A deterministic CPD provides a conservative, discretized approximation. The randomized version yields a continuous distribution where the probability integral transform of the true label is stochastically larger than a uniform distribution, a property known as super-uniformity.

Full predictive distributions enable risk-aware decision-making beyond simple set membership. Use cases include:

• Inventory management: Setting reorder points based on the full demand distribution, not just a prediction interval
• Medical dosing: Choosing a dose that balances efficacy and toxicity risk using the conditional CDF of patient response
• Financial reserves: Calculating value-at-risk (VaR) and expected shortfall directly from the calibrated distribution

The core heuristic engine driving all conformal methods. It quantifies how unusual a specific input-label pair appears relative to a calibration dataset. In the context of CPDs, the choice of nonconformity measure—such as the absolute residual from a conditional density estimator—directly shapes the resulting predictive distribution. A well-designed measure is critical for achieving informative and adaptive distributions.

A direct precursor to CPDs that wraps a standard quantile regression model with a conformal calibration step. This corrects the predicted quantile intervals to achieve finite-sample validity. CPDs generalize this idea by calibrating the entire conditional distribution, not just a few discrete quantiles, to produce a smooth cumulative distribution function.

A specialized application integrating conformal prediction with survival models to produce lower prediction bounds for survival times with guaranteed coverage. This directly addresses the challenge of right-censored data. CPDs extend this by providing a complete predictive distribution over the survival time, not just a lower bound, offering a full picture of time-to-event uncertainty.

Applies conformal prediction to estimate valid confidence intervals for individual treatment effects (ITEs). This provides distribution-free uncertainty quantification for counterfactual predictions. CPDs advance this by generating a full predictive distribution for the potential outcome under each treatment, enabling richer decision-making in personalized policy design.

A stronger validity objective that seeks to guarantee coverage for specific subgroups or feature values. While exact conditional coverage is statistically impossible without assumptions, CPDs aim to approximate it by modeling the entire conditional distribution. This makes them a powerful tool for achieving approximate conditional validity, ensuring reliable uncertainty estimates across heterogeneous populations.