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Methods

While loss functions define what your model optimizes, the methods in this section define how to extract reliable uncertainty, correct for data issues, and validate your results.

Use the taxonomy below to find the right method for your task, or read each section in order — the flow goes from most commonly needed (conformal, ensembles) to more specialized tools (causal inference, PPI).

Three questions every method answers

Beyond the loss function, a regression method is defined by which part of the predictive distribution it identifies and what it guarantees:

  1. What is the forecast object? A point \(\hat{y}(x)\), an interval \([L(x), U(x)]\), a density \(p_\theta(y \mid x)\), or a set of members \(\{y^{(m)}(x)\}\).
  2. Where does the uncertainty come from? Aleatoric noise is the conditional spread of \(Y\) given \(x\); epistemic uncertainty is uncertainty about the model itself. Their sum is the predictive variance $$ \operatorname{Var}(Y \mid x, \mathcal{D}) = \underbrace{\mathbb{E}{\theta \mid \mathcal{D}}\big[\operatorname{Var}(Y \mid x, \theta)\big]}}} + \underbrace{\operatorname{Var{\theta \mid \mathcal{D}}\big[\mathbb{E}(Y \mid x, \theta)\big]}. $$}
  3. What is the guarantee? Some methods are purely empirical (a better-trained variance head); others carry a finite-sample statement, e.g. $$ P\big(Y_{n+1} \in \hat{C}(X_{n+1})\big) \ge 1-\alpha $$ for conformal prediction under exchangeability, or asymptotic normality for the doubly-robust causal estimators.

Keeping those three questions separate prevents the most common category error in practice: treating a coverage guarantee as an epistemic decomposition, or treating ensemble disagreement as a calibrated density. The uncertainty-decomposition guide states exactly which contracts each family does and does not satisfy.

Method Taxonomy

graph LR
    A["Your Task"] --> B{"Need coverage<br/>guarantees?"}
    B -->|"Yes"| C["Conformal Prediction"]
    B -->|"No"| D{"Data quality<br/>issues?"}
    D -->|"Noisy inputs"| E["EIV / RC / SIMEX"]
    D -->|"Distribution shift"| F["Test-Time Adaptation"]
    D -->|"Clean"| G{"Output shape?"}
    G -->|"Point + σ"| H["Ensemble + Calibration"]
    G -->|"Multimodal"| I["MDN / Flows"]
    G -->|"Intervals"| J["Quantile + Conformal"]

    click C "conformal/index.md"
    click E "algorithms/rc.md"
    click F "test-time/ot-shift-conformal.md"
    click H "ensemble/index.md"

Sections

Conformal Prediction

Distribution-free prediction intervals with finite-sample coverage guarantees. Wrap any pre-trained model for calibrated intervals — Split conformal (fast baseline), CQR (heteroscedastic), CTI (multimodal), and density-aware variants. Start here if you need coverage guarantees.

Ensembles for Uncertainty

Deep Ensembles, BatchEnsemble, SWAG, MC-Dropout, BNN — epistemic uncertainty via member disagreement; aleatoric decomposition when members predict variance (e.g. HeteroscedasticEnsembleModel). Start here if you need to know what the model doesn't know.

Algorithms

Specialized training algorithms: IRLS for robust fitting, RC and SIMEX for measurement error correction, TIC-TAC for covariance learning, Heteroscedastic Laplace for last-layer Bayesian regression, and Adaptive Prior VI for covariate-shift robustness, VIR for imbalanced targets with evidential uncertainty, and Continuous Group DRO for worst-regime robustness. Start here if standard training isn't enough for your data.

Test-Time & Shift

Adapt models at deployment without retraining: Bayesian linear heads for closed-form posterior updates, OT-based conformal reweighting for covariate shift, and shift-factored predictive transport. Start here if your test distribution differs from training.

Tabular MLP

A strong-default neural backbone for tabular data (robust scaling, periodic embeddings, AdamW with early stopping) that works with any torchregress loss head. Start here if you need a well-tuned network without a tuning loop.

Model-Agnostic UQ Wrappers

Conformal intervals (split, normalized, CQR, CV+, Jackknife+) and variance calibration on top of any scikit-learn, LightGBM, torch or TabularFit base, plus a one-call calibrated deep-ensemble recipe. Experimental in 0.3, may change in 0.4. Start here if you already have a good point model and need intervals.

Post-Hoc Calibration

Improve uncertainty quality after training: variance temperature scaling, isotonic mean calibration, PIT calibration. One-line API, no retraining needed. Always calibrate before trusting your intervals.

Constraints

Enforce structural properties on model outputs: non-negativity, bounded ranges, simplex, monotonic quantile curves, Lipschitz bounds. Drop-in nn.Module wrappers. Start here if your domain requires hard output constraints.

Causal Inference

Doubly-robust ATE / CATE estimation with cross-fitting and overlap diagnostics. Start here if you need treatment-effect estimates under confounding.

Prediction-Powered Inference

Confidence intervals under limited labels — combine a small trusted-labeled set with a large model-predicted pool for efficient statistical inference. Start here if you have abundant unlabeled data but scarce labels.

Semi-Supervised Regression

Teacher–student consistency with distributional pseudo-labels: an EMA teacher builds a consensus predictive batch on unlabeled inputs, and the student is penalised for disagreeing with it. Start here if labels are scarce but unlabeled inputs are abundant.

Flow Matching & Normalizing Flow Maps

Simulation-free conditional generation with a learned velocity field (flow matching) and two-time invertible pushforwards with exact endpoint likelihoods (flow maps). Start here when you need multimodal conditional samples rather than a parametric density.

WALDO Wald Regions

Simulator-free confidence regions for regression parameters from predictive mean and covariance, with a critical-value regressor calibrated on held-out data. Start here when you want parameter confidence sets rather than prediction intervals.

Visualization

Diagnostic plots for calibration, residuals, learning curves, and method comparisons. Start here when you want to visually validate your model's behavior.

Next steps