Two-Stage Conformal Prediction Framework

• The paper presents a two-stage framework that leverages a fast, preliminary prediction to trim candidate outcomes followed by a precise, slow model to refine prediction intervals.
• It improves computational efficiency by limiting intensive recalculations to a smaller subset while nearly preserving ideal finite-sample coverage.
• The method incorporates residual decomposition for stagewise uncertainty attribution and adaptive calibration, aiding diagnostics under distribution shifts.

A conformal prediction framework for two-stage sequential models addresses the challenge of constructing statistically rigorous uncertainty sets (prediction intervals or sets) when predictions are obtained in two modular steps—typically, an upstream model generates an intermediate representation, which is subsequently used by a downstream model for the final prediction. This sequential approach is prevalent in high-dimensional regression, sparse inference, risk control in ranking or detection, and modular decision pipelines. The theoretical and empirical literature on multi-stage or modular conformal prediction demonstrates that such frameworks can provide stronger computational efficiency, finer uncertainty attribution, and improved diagnostics compared to monolithic approaches, while preserving (at least approximately) the strong finite-sample coverage properties central to conformal prediction.

1. Sequential Structure and Motivation

Two-stage sequential models divide the predictive process into two explicit components. The upstream stage produces an intermediate output or a coarse prediction—such as feature extraction, candidate screening, or a trimming operation to remove unlikely values. The downstream (second) stage uses this representation—often with a more accurate but computationally expensive model—to make the final prediction or to refine the prediction set.

Trimmed Conformal Prediction (TCP) (Chen et al., 2016) is a canonical approach: a "fast but less accurate" regression model is first used to quickly exclude large regions of response space ("trimming step"), creating a trial set $T$ with high probability of containing the true value. Then, a "slow but accurate" method—often computationally intensive or limited to complex models (e.g., the lasso in high-dimensions)—is applied only to $T$ ("prediction step"), constructing the final conformal set $C$. This two-stage structure is mirrored in risk-control for ranking (Xu et al., 27 Apr 2024), modular pipeline handling (Zhang et al., 6 Oct 2025), and clinical uncertainty estimation with zero-inflated outcomes (Diaz-Rincon et al., 14 Aug 2025).

The sequential structure enables principled resource allocation, attributes uncertainty to specific stages, and increases flexibility for handling high-dimensional, modular, or nonstationary data.

2. Formal Two-Stage Conformal Prediction Procedures

TCP: Trimmed Conformal Prediction

Let $n$ labeled observations $(X_1,Y_1),\ldots,(X_n,Y_n)$ be given, and for a new input $X_{n+1}$, the task is to predict $Y_{n+1}$.

1. Trimming Step: Choose a fast (possibly less accurate) regression model $\mu_y^{(\text{fast})}$.
   • Compute the trial set

   $$T = \left\{ y \in \mathbb{R} : |(r^{(\text{fast})}_y)_{n+1}| \text{ is in the bottom } (1-\alpha_{\text{trim}}) \text{ quantile of } \left\{ |(r^{(\text{fast})}_y)_i| \right\}_{i=1}^{n+1} \right\}$$

   where the residuals $r^{(\text{fast})}_y$ are computed by including the candidate $y$ as $Y_{n+1}$.

2. Prediction Step: Use a slow but accurate model $\mu_y^{(\text{slow})}$ to form the set

   $$C = \left\{ y \in T : |(r^{(\text{slow})}_y)_{n+1}| \text{ is in the bottom } (1-\alpha_{\text{pred}}) \text{ quantile of } \left\{ |(r^{(\text{slow})}_y)_i| \right\}_{i=1}^{n+1} \right\}.$$

   The overall guarantee is

   $$\mathbb{P}(Y_{n+1} \in C) \geq 1 - (\alpha_{\text{trim}} + \alpha_{\text{pred}})$$

   By choosing $\alpha_{\text{trim}}$ small, the loss of coverage relative to ideal conformal prediction is negligible; computation is reduced by restricting slow model refits to the trimmed set.

Residual Decomposition in Modular Pipelines

For two-stage models where $$w \stackrel{\hat{\mu}_1}{\rightarrow} x \stackrel{\hat{\mu}_2}{\rightarrow} y$$ (Zhang et al., 6 Oct 2025):

• Compute the end-to-end residual $R = |y - \hat{\mu}_2(\hat{\mu}_1(w))|$

• Decompose into stagewise components:

  • Downstream residual: $R_2 = |y - \hat{\mu}_2(x)|$
  • Upstream delta: $$\Delta R_1 = |R_2 - |y - \hat{\mu}_2(\hat{\mu}_1(w))||$$

Then,

$R \leq \Delta R_1 + R_2$

Prediction intervals are then constructed as:

$$\hat{C}_{a,b,\alpha}(w_{\text{test}}) = \hat{\mu}_2(\hat{\mu}_1(w_{\text{test}})) \pm [a \cdot Q_{1-\alpha}(\Delta R_1) + b \cdot Q_{1-\alpha}(R_2)]$$

where $Q_{1-\alpha}(\cdot)$ denotes the empirical $1-\alpha$ quantile, and $a,b$ are scale parameters selected by risk-controlled calibration.

This decomposition allows practitioners to attribute the predictive uncertainty to each stage—enabling stage-targeted diagnostics and robust uncertainty quantification under distribution shift.

Calibration for Stagewise Parameters

Validated coverage is assured by calibrating the interval via family-wise error rate (FWER) control over a candidate grid of $(a, b)$ (or quantile levels), identifying all parameter pairs for which empirical miscoverage on a calibration set does not exceed $\alpha$. This empirically guarantees, with high probability, that the interval achieves the desired level of coverage for any validated $(a, b)$ pair.

An adaptive extension updates the calibration window and scaling/quantile parameters online (using a sliding window and FWER re-selection) for nonstationary environments, ensuring long-run average coverage converges to $1 - \alpha$.

3. Computational Advantages and Statistical Guarantees

The principal motivation for two-stage conformal prediction is computational efficiency:

• Trimming in TCP reduces the computational expense of conformal prediction in high-dimensional regression by restricting the computationally intensive refits to a small region containing the probable true value.
• By using the full data in the second stage (unlike split conformal prediction), methods such as TCP retain sharper prediction intervals and better statistical efficiency at nearly the same coverage.

The modular residual decomposition framework (Zhang et al., 6 Oct 2025) further enables the separate calibration and attribution of uncertainty, so intervals remain valid even in the face of stage-specific distributional shifts that degrade standard conformal coverage.

The union bound, FWER control, and adaptive re-calibration methods ensure that the coverage of final prediction intervals meets or exceeds the nominal level in finite samples, provided standard conformal or monotonicity requirements are met in the construction.

4. Applications to High-Dimensional, Sparse, and Modular Pipelines

TCP and related two-stage schemes have demonstrated broad utility:

• Sparse regression: When employing the lasso (with high-dimensional, sparse data), re-fitting for every candidate outcome is computationally prohibitive. TCP uses ridge regression (with closed-form residuals $r_y^\text{(ridge)}=u+vy$) or split conformal trimming with the lasso, so that final conformal sets can be efficiently computed only over the trimmed region (Chen et al., 2016).
• Supply chain and finance: Residual decomposition and stagewise calibration handle dynamic supply chain predictions and stock market data, maintaining coverage under shifting data distributions, and offering interpretability via stage-specific uncertainty attribution (Zhang et al., 6 Oct 2025).
• Clinical prediction: In zero-inflated scenarios (e.g., forecasting Parkinson's Disease medication changes), a first-stage classifier identifies probable no-change cases. The conformal interval is applied only to instances likely to undergo change, yielding reduced prediction interval widths while maintaining marginal coverage (Diaz-Rincon et al., 14 Aug 2025).

Key results from empirical studies confirm that:

• TCP achieves as sharp or sharper prediction intervals as full-data conformal lasso, but with vastly fewer evaluations needed;
• Stagewise modular methods retain coverage even when simulated distribution shifts affect only a single step of the pipeline, whereas standard conformal approaches lose coverage in such settings.

5. Diagnostic and Interpretability Benefits

A major advantage of two-stage decomposition is the ability to attribute prediction interval width (and lack of coverage) to individual model components:

• If $\Delta R_1$ (upstream error) increases after an upstream data distribution shift, this signals the need for upstream retraining;
• If $R_2$ (downstream error) is large under stable upstream conditions, the downstream model is the bottleneck.

This interpretability is not accessible in black-box conformal prediction, where error attribution is aggregated and can only be inferred indirectly. Real-time tracking of the evolution of quantile levels and scale parameters further enables online diagnostics for industrial pipelines.

6. Limitations and Future Directions

While two-stage frameworks demonstrate strong coverage and computational advantages, several limitations are noted:

• The coverage guarantee can be slightly more conservative than the ideal (loss of up to $\alpha_\text{trim}$ in TCP), although this can be made negligible with careful parameter selection.
• Risk attributions depend on the pipeline decomposition and monotonicity of residual error propagation.
• Extensions to more than two stages, highly heterogeneous model families, or non-exchangeable settings (e.g., adversarial or time-series data) require additional theoretical development.
• Real-world impact is sensitive to the choice of residual quantiles, calibration splitting, and appropriate tuning of FWER control parameters in modular pipelines.

7. Table: Summary of Key Two-Stage Conformal Prediction Approaches

Framework	Stage 1 (Upstream)	Stage 2 (Downstream)	Coverage Guarantee
TCP (Chen et al., 2016)	Trimming (fast model, e.g., ridge or split lasso)	Accurate model (e.g., lasso, slow conformal)	$1-(\alpha_\text{trim}+\alpha_\text{pred})$
Residual Decomposition (Zhang et al., 6 Oct 2025)	Intermediate representation prediction ($\mu_1$)	Final prediction from upstream output ($\mu_2$)	Empirical FWER-calibrated near $1-\alpha$
Clinical zero-inflated (Diaz-Rincon et al., 14 Aug 2025)	Change/no-change classifier	Regression conformal interval (if actionable)	Empirical marginal (1–α) coverage

The explicit leveraging of modular structure, residual decomposition, stagewise quantile calibration, and FWER-based risk control defines the modern conformal prediction framework for two-stage sequential models. These innovations combine computational tractability with theoretical validity, while offering indispensable transparency for diagnostic and robust sequential decision-making.