Distribution Forecasts Overview

• Distribution forecasts are predictive outputs that provide complete probability distributions instead of single estimates, enabling direct extraction of risk metrics.
• They employ methods such as parametric, semi-parametric, copula, and deep learning techniques to tailor forecasts for diverse applications including energy, epidemiology, and finance.
• Robust calibration, ensemble aggregation, and evaluation with scoring rules like CRPS and log score ensure reliable uncertainty quantification and decision support.

A distribution forecast is a predictive output that provides a full probability distribution—or a functionally precise representation thereof—for a future or latent quantity of interest, rather than a single point estimate. Distributional forecasting produces outcomes such as predictive densities, quantile sets, cumulative distribution functions (CDFs), or functionally equivalent representations, enabling quantification of forecast uncertainty and direct extraction of any desired risk metric (e.g., specific quantiles, value at risk, prediction intervals). Distribution forecasts are foundational in modern statistics, machine learning, time series, energy markets, epidemiology, and actuarial applications, where knowledge of the full conditional distribution is crucial for robust decision support, risk management, and scenario generation.

1. Fundamental Principles of Distribution Forecasts

A distribution forecast for a target variable $Y$ given information $\mathcal{F}_t$ is a predictive law $F_{Y|\mathcal{F}_t}$, characterized by its cumulative distribution function (CDF) $F(y) = P(Y \leq y \mid \mathcal{F}_t)$, or equivalently by a predictive density $f(y)$ if it exists. This approach enables the derivation of any probabilistic statistic—mean, quantiles, predictive intervals, expected shortfall—directly from the forecast output, as opposed to point forecasting, which provides only a single central value (e.g., conditional mean or median), discarding all uncertainty information.

Distribution forecasting readily supports scenario generation (sampling), risk analysis (tail estimates), and calibration diagnostics, and is rigorously evaluated using strictly proper scoring rules such as the log score, continuous ranked probability score (CRPS), and quantile (pinball) loss (Marcjasz et al., 2022, Austnes et al., 2023, Schulz et al., 2022, Wang et al., 2022).

2. Core Methodological Families

Distribution forecasting methods can be classified into several methodological families, each providing a distinct representation of the predictive distribution and entailing different modeling, estimation, and validation regimes.

a) Parametric and Semi-Parametric Forecasts

Parametric methods postulate a specific family of distributions (e.g., Normal, Johnson's SU, skew-t), parameterized by $\theta$, and estimate these parameters from data or features. For example, in electricity price forecasting, a distributional deep neural network ("DDNN") produces, for each time-step, either a Normal law (location-scale) or a Johnson SU law (with additional skewness and kurtosis parameters) (Marcjasz et al., 2022, Hirsch et al., 2022). Semi-parametric variants relax parametric constraints by modeling quantile functions or CDFs directly (e.g., Bernstein quantile networks, histogram estimation networks) (Schulz et al., 2022).

b) Copula and Nonparametric Models

Empirical copula-based techniques construct a nonparametric estimate of the joint distribution of target and covariates on the unit hypercube (via pseudo-observations), then transform back to the original scale to deliver predictive conditionals for the variable of interest. For instance, in load forecasting, empirical copula density estimation with beta kernels on rank-transformed data facilitates extraction of quantiles under nonlinear dependence and boundary-adaptive behavior (Austnes et al., 2023). Bandwidth selection is performed via integrated square error minimization.

c) Deep Learning and Ensemble Architectures

Modern approaches leverage deep neural networks to output full predictive distributions, either through distributional regression (directly estimating family parameters), quantile regression layers, or histogram probability outputs. Deep ensembles—collections of independently initialized DNNs—require distribution forecast aggregation; strategies include probability pooling (linear pool), quantile (Vincentization), and newer hybrid approaches such as angular combining (interpolating between probability- and quantile-averaging to optimize sharpness versus coverage) (Schulz et al., 2022, Taylor et al., 2023).

d) Mixture and Empirical Bootstrapping Methods

In collaborative and robust settings, forecasts are represented as finite (discrete) mixtures: $F(z) = \sum_k w_k F_k(z)$, where each $F_k$ stems from a model, a sample-based empirical law, or a bin/quantile-based construction (Wadsworth et al., 2023). Mixture-format forecasts can be efficiently scored, ensembled, and allow storage/performance advantages over raw draws or quantile sets. Nonparametric bootstrap methods generate empirical predictive distributions by aggregating residuals from time-preserving backtests, with adaptive selection mechanisms to condition on forecast meta-information (Wang et al., 2022).

3. Distribution Forecast Construction: Model-Specific Illustrations

Different modeling domains and problem structures motivate tailored construction of distribution forecasts:

• Electricity Market Prices: A DDNN outputs parametric distributions for each hour's price, capturing spikes and heavy tails by dynamically adapting skewness and kurtosis (via Johnson's SU parameters). Training optimizes the sum of negative log-likelihoods; evaluation uses log score and CRPS. For risk management tasks (VaR, ES), accurate tail modeling yields robust reserve sizing and risk limits (Marcjasz et al., 2022, Hirsch et al., 2022).
• Load Forecasting: Empirical copula approaches flexibly accommodate variables with disparate marginal distributions; full predictive distributions of load condition on weather and temporal covariates, enabling quantile-based scenarios and outperforming quantile regression in CRPS and pinball loss (Austnes et al., 2023). Global modeling with deep neural networks leverages sharing across spatial or hierarchical units, with fine-tuning and ensembles for localization (Grabner et al., 2022).
• Hierarchical Forecasts and Reconciliation: In multilevel structures (e.g., utility networks, multi-product demand), probabilistic reconciliation ensures that base distribution forecasts at different aggregation levels are rendered coherent by conditioning on hierarchical constraints. For Gaussian base laws, reconciliation is analytic (via Schur complement/minT). For count data, importance weighting and empirical assessment yield reconciled (possibly non-shrinking) predictive variance (Corani et al., 2019, Zambon et al., 2023, Nespoli et al., 2019).
• Non-Stationary and Collaborative Environments: For epidemics and other collaborative forecasting settings, distribution recalibration post-processing (PIT-based mapping, black-box transformations) achieves in-sample improvement in calibration and log score, independent of base model specifics (Rumack et al., 2021). Ensemble creation via mixture distribution format and weight optimization permits rigorous probabilistic scoring, storage efficiency, and real-time adaptability (Wadsworth et al., 2023).

4. Aggregation and Combination of Distribution Forecasts

When multiple distribution forecasts are available for the same target, combination methods are utilized:

• Linear Pool (Probability Pooling): The average of component CDFs, ensuring overdispersion but sometimes excessively wide predictive intervals.
• Quantile Pool (Vincentization): The average (possibly weighted and/or biased-shifted) of quantile functions, preserving distributional family and often delivering sharper, well-calibrated aggregates. Estimation of intercept and weights via cross-validated CRPS minimization is standard practice.
• Angular Combining: Interpolates between probability and quantile pooling (parameterized by angle θ), with θ chosen to optimize scoring rules (e.g., CRPS). This method ensures mean preservation, increases variance with θ, and is theoretically guaranteed to at least match average component performance in proper scores (Taylor et al., 2023). Empirical evidence supports its superiority in several practical applications, including COVID-19 and electricity price forecasting.

A summary of aggregation schemes:

Method	Pooling Scale	Dispersion
Linear Pool	Probability (vertical)	Overdispersion
Quantile Pool	Quantile (horizontal)	Underdispersion/Sharp
Angular Combining	Hybrid	Tunable (θ)

5. Calibration, Evaluation, and Diagnostics

Distribution forecasts are evaluated and calibrated via strictly proper scoring rules:

• Logarithmic Score: Negative log density at the realized value.
• Continuous Ranked Probability Score (CRPS): Integrated squared CDF error, or equivalently average pinball loss over quantiles.
• Pinball Loss: Weighted absolute error by quantile level; used for quantile predictions.
• Interval and Energy Scores: Assess predictive interval width and multivariate forecast sharpness and coverage.

Calibration is commonly assessed via Probability Integral Transform (PIT) histograms, coverage metrics, and diagnostic plots such as threshold-martingale processes (e.g., for evolving sequences), with recalibration algorithms available to correct systematic deviations (Rumack et al., 2021, Foster et al., 2021).

6. Domain Applications and Decision-Theoretic Implications

Distribution forecasts underpin key operational and policy decisions:

• Risk Management: Accurate modeling of predictive tails via distribution forecasts is critical for VaR, ES, and hedging against price or load spikes (Marcjasz et al., 2022, Hirsch et al., 2022).
• Stochastic Programming: In energy markets and supply chains, the appropriate complexity of the distribution forecast (expectation, marginal distributions, bivariate or joint laws) closely matches the structure of the underlying optimization problem, with higher-order dependencies required for recourse policies involving startup costs, storage coupling, or nonlinear constraints (Beykirch et al., 2022).
• Annuity and Survival Analysis: Forecasts of full age-at-death distributions directly drive life expectancy and annuity value calculations, with compositional data transformation frameworks yielding accurate point and interval projections for actuarial purposes (Shang et al., 6 Jul 2025).
• Grid Operation under Uncertainty: In power systems with high renewable penetration, probabilistic load and PV forecasts (e.g., CGAN-based) feed directly into chance-constrained optimal power flow formulations for the computation of operating envelopes that respect system security at prescribed risk levels (Yi et al., 2022).

7. Practical Recommendations and Limitations

• Model Selection and Aggregation: Use the simplest forecast form just sufficient for the targeted decision problem and data regime. Regularization, ensemble aggregation, and calibration post-processing are essential for robust distribution forecast accuracy (Beykirch et al., 2022, Schulz et al., 2022).
• Nonparametric and Data-Driven Approaches: Nonparametric empirical and copula-based methods avoid restrictive assumptions and can capture nonlinearities or dependence structures inadequately addressed by parameteric models (Austnes et al., 2023, Wang et al., 2022).
• Ensemble and Combination Methods: For deep learning and collaborative forecasting, quantile scaling (Vincentization or angular combining) is generally preferred to linear pooling. Ensemble size ≥10 suffices in most applications; further gains are minimal beyond n=20 (Schulz et al., 2022, Taylor et al., 2023).
• Calibration and Performance Monitoring: Routinely apply PIT-based calibration diagnostics and recalibration or volatility correction methods to maintain reliable uncertainty quantification. Distribution recalibration and threshold-martingale filtering are effective black-box post-processing steps (Rumack et al., 2021, Foster et al., 2021).
• Limitations: Nonparametric smoothing may overfit with limited data; structured regression models can miss regime shifts; mixture and ensemble model weights may require regular retraining (Rumack et al., 2021, Wang et al., 2022). Tail calibration and sharpness can be especially challenging at endpoints or under severe nonstationarities (Taylor et al., 2023, Wadsworth et al., 2023).

Distribution forecasts constitute the rigorous foundation for modern risk-aware decision-making in complex, uncertain environments across scientific, financial, actuarial, and engineering domains. Their construction, calibration, aggregation, and deployment are guided by a spectrum of advanced statistical, machine learning, and domain-specific procedures, readily adapting to emerging data modalities and operational requirements.