Conformal Prediction

The foundational requirement for valid conformal inference. A sequence of random variables is exchangeable if their joint distribution is invariant to permutation. In financial contexts, this means the calibration set and each test point must be drawn from the same underlying regime. Key implications:

• Weaker than the i.i.d. assumption—allows for certain dependencies
• Violated during regime shifts (e.g., a volatility spike), requiring retraining or adaptive weighting
• Enables the rank-based construction of prediction intervals without knowing the true data distribution
• For time series, conformal prediction can be extended via rolling calibration windows to approximate exchangeability

A real-valued function that quantifies how unusual a new example is relative to a calibration set. The choice of nonconformity measure directly shapes the resulting prediction intervals. Common measures:

• Absolute residual: |y - ŷ| for regression—produces constant-width intervals
• Normalized residual: |y - ŷ| / σ̂(x) where σ̂(x) is an estimated local variability—produces adaptive intervals that widen in high-uncertainty regions
• Quantile-based: max{q̂_low(x) - y, y - q̂_high(x)} for conformalized quantile regression
• The measure must be symmetric in its arguments to ensure valid marginal coverage

Split conformal prediction partitions the available data into a proper training set and a disjoint calibration set. This avoids the computational cost of full conformal prediction while maintaining validity. The process:

• Train the underlying model f̂ on the training set only
• Compute nonconformity scores for all points in the calibration set
• For a target coverage of 1 - α, find the ⌈(n+1)(1-α)⌉ / n quantile of calibration scores
• The prediction interval for a new point x_new is: [f̂(x_new) - q̂, f̂(x_new) + q̂] (for absolute residual measure)
• Finite-sample guarantee: P(Y_test ∈ C(X_test)) ≥ 1 - α holds exactly, not asymptotically

A powerful extension that combines quantile regression with conformal calibration to produce adaptive prediction intervals that naturally widen in volatile market conditions. How it works:

• Train a quantile regressor to output lower (q̂_α/2) and upper (q̂_1-α/2) conditional quantiles
• Compute nonconformity scores on the calibration set: s_i = max{q̂_α/2(x_i) - y_i, y_i - q̂_1-α/2(x_i)}
• Adjust both quantiles by the calibration quantile Q_{1-α} of these scores
• Result: Intervals that are short in calm markets and wide during high-volatility events, without sacrificing the coverage guarantee
• Particularly valuable for Value-at-Risk (VaR) estimation and options pricing

A critical distinction in conformal prediction that affects practical utility. Marginal coverage guarantees that intervals cover the true value on average across all test points—the standard conformal guarantee. Conditional coverage would require coverage for every specific X = x, which is impossible without distributional assumptions. Practical implications:

• Marginal coverage may over-cover in low-volatility regimes and under-cover in high-volatility ones
• Mondrian conformal prediction achieves approximate conditional coverage by partitioning the feature space into categories and calibrating separately within each
• For trading, partitioning by volatility regime or market cap improves practical reliability
• The trade-off: finer partitions reduce calibration set size per bin, increasing interval width

Standard conformal prediction assumes exchangeability, which is violated by the serial dependence in financial time series. Adaptive conformal inference (ACI) and related methods address this:

• Rolling calibration windows: Use only the most recent w observations as the calibration set, discarding older data that may reflect a different regime
• Adaptive Conformal Inference: Dynamically adjusts the effective coverage level α_t based on recent miscoverage frequency using an online update rule: α_{t+1} = α_t + γ(α - err_t)
• EnbPI (Ensemble Batch Prediction Intervals): Uses bootstrap ensembles and a leave-one-out style calibration for time series with serial correlation
• These methods provide approximate validity under weak dependence assumptions, making them suitable for high-frequency forecasting

An interval estimate for a future observation that, unlike a confidence interval (which bounds a population parameter), bounds the actual value of a single random variable with a specified probability. Conformal prediction produces distribution-free prediction intervals with finite-sample validity. Key properties:

• Marginal coverage: Guarantees that P(Y_{n+1} ∈ C(X_{n+1})) ≥ 1 − α on average over calibration and test points
• Conditional coverage: A stronger, often unattainable goal requiring coverage for each specific X = x
• Adaptive intervals: Width varies with input difficulty—wider in high-uncertainty regions, narrower where the model is confident

A real-valued function A(z, S) that quantifies how unusual or nonconforming a data point z is relative to a multiset S. The choice of nonconformity measure determines the adaptivity and efficiency of the resulting prediction intervals. Common choices:

• Absolute residual: A(z, S) = |y − μ̂(x)|, where μ̂ is a point predictor trained on S
• Normalized residual: A(z, S) = |y − μ̂(x)| / σ̂(x), using an auxiliary model σ̂ to estimate local variability, producing locally adaptive intervals
• Quantile-based: A(z, S) = max{q̂_α(x) − y, y − q̂_{1−α}(x)}, leveraging quantile regression outputs

A held-out dataset, disjoint from the training set, used exclusively to compute nonconformity scores for calibrating the conformal predictor. The split-conformal method partitions data into:

1. Proper training set: Used to fit the underlying model μ̂
2. Calibration set: Used to compute empirical quantiles of nonconformity scores
3. Test point: The new observation for which a prediction interval is constructed The size of the calibration set directly affects interval stability—larger sets reduce variance in the empirical quantile estimate. For a target coverage of 1−α with n calibration points, the finite-sample coverage guarantee is at least 1−α, with an upper bound of 1−α + 1/(n+1).

A statistical technique that estimates the conditional quantiles of a response variable, forming the basis for Conformalized Quantile Regression (CQR). Unlike mean regression, quantile regression directly models:

• Lower quantile q̂_{α/2}(x): The value below which α/2 of observations fall
• Upper quantile q̂_{1−α/2}(x): The value below which 1−α/2 of observations fall CQR wraps these initial quantile estimates with a conformal correction term, adjusting the interval width to achieve valid finite-sample coverage even when the underlying quantile regressor is mis-specified. This is particularly valuable in high-frequency trading where return distributions exhibit heavy tails and heteroskedasticity.

The probabilistic assurance that a conformal prediction interval contains the true value with at least the specified confidence level. Two types of validity:

• Marginal validity: E[1_{Y_{n+1} ∈ C(X_{n+1})}] ≥ 1 − α, holding on average across all possible calibration and test draws
• Conditional validity: P(Y_{n+1} ∈ C(X_{n+1}) | X_{n+1} = x) ≥ 1 − α for almost all x—impossible to achieve with finite samples without distributional assumptions In practice, cross-conformal and jackknife+ methods improve computational efficiency while maintaining approximate validity, though exact finite-sample guarantees are sacrificed for reduced model retraining costs.