Retrospective Forecasting (RF) Analysis

• Retrospective Forecasting (RF) is a systematic method for evaluating prediction models using only the data available at the original forecast time.
• RF employs rolling retraining, strict temporal splits, and diverse methodologies like ARIMA, GAM, and LLM-based approaches to measure forecast performance.
• This approach underpins reliable benchmarking in areas such as epidemic analytics and language model forecasting by providing actionable insights into model calibration and accuracy.

Retrospective forecasting (RF) is the systematic evaluation of forecasting models by applying them to historical events whose outcomes are already known, with the constraint that only information available at the time of the original forecast origin is used. RF is central to model benchmarking in both epidemic analytics and LLM forecasting, enabling direct measurement of out-of-sample forecast skill, calibration, and comparative model performance in environments where prospective evaluation would entail prohibitive latency. RF’s reliability, methodological structure, and limitations are contingent on the particular use case—statistical models with strict temporal splits differ fundamentally from LLMs trained on internet-scale corpora that may include resolved event data.

1. Formal Definition and Theoretical Foundation

Retrospective forecasting (RF) involves generating forecasts for past events using only the data that would have been available at the event’s forecast origin, thereby emulating a prospective forecasting scenario in a controlled post hoc setting. In epidemic modeling, RF permits the sequential retraining and out-of-sample projection of statistical and mechanistic models, facilitating robust performance comparison and recalibration. In the context of LLMs, RF additionally confronts unique epistemic constraints due to ever-advancing model knowledge cutoffs. Key formalism includes:

• For each forecast origin $t$, RF assembles input data up to $t$ and generates forecast targets for $t+h$, $h$ in the desired horizon.
• Models are refit or re-run at each origin to prevent leakage from future data.
• In LLMs, RF operationalizes either True Ignorance (TI) or Simulated Ignorance (SI):
  • TI: Event resolution $R(q)$ postdates model cutoff $K(m)$; model cannot have seen the true outcome.
  • SI: Model is prompted to ignore any post-cutoff knowledge $C$ for pre-cutoff events, despite including them in training (Li et al., 20 Jan 2026).

This foundational rigor is essential for real-world validation and for calibrating operational deployment of forecasting systems (Alhassan et al., 30 Nov 2025).

2. Methodological Protocols in Retrospective Forecasting

The principal RF workflow comprises:

• Selection of historical data windows that emulate strict temporal separation (no future leakage).
• Rolling retraining or re-estimation of model parameters using only pre-origin data.
• Application of multiple models (statistical, ML, phenomenological, LLM-based) to shared historical questions or targets.
• Generation of probabilistic forecasts and confidence intervals via parametric bootstrapping or analytic uncertainty quantification.
• Tabulation of forecast-observation pairs for performance metric computation.

In epidemic modeling (e.g., COVID-19 wastewater surveillance (Alhassan et al., 30 Nov 2025)), RF was executed weekly from March 2022 to September 2024 at five spatial scales, creating ≈10,000 out-of-sample forecasts. At each weekly origin $t$:

• 10 previous weeks’ data assembled: $[t-9, ..., t]$.
• All model types refit under Gaussian error assumption.
• Forecasts generated for $t+1$ … $t+4$ weeks.
• Bootstrapped medians and 95% prediction intervals computed.
• Iterative advancement in $t$ to cover retrospective time series.

No future leakage was permitted, preserving the integrity of error estimation and model comparison.

3. Model Classes and Formulations in RF Practice

A range of models are commonly employed in RF studies:

• ARIMA (AutoRegressive Integrated Moving Average): $$\phi(B)\,(1 - B)^d\,y_t = c + \theta(B)\,\epsilon_t$$; optimal parameters chosen via Hyndman–Khandakar algorithm.
• Generalized Additive Model (GAM): $y_t = \beta_0 + s(t) + \epsilon_t$, where $s(t)$ is a penalized spline fit.
• Simple Linear Regression (SLR): $y_t = \beta_0 + \beta_1 t + \epsilon_t$.
• Prophet: $y(t) = g(t) + s(t) + h(t) + \epsilon_t$, combining trends, seasonality, and holiday effects.
• n-Sub-epidemic Framework: $$\frac{dC_i(t)}{dt} = A_i(t) r_i C_i(t)^{p_i} (1 - \frac{C_i(t)}{K_{0,i}})$$, with ensemble construction via Akaike weights.

Ensemble models (weighted and unweighted, e.g., EM 2 UW, EM 3 UW) are frequently used to hedge against model misspecification and improve calibration. LLM-based forecasting employs direct and chain-of-thought prompt templates, with performance stratified by knowledge cutoff conditions (Li et al., 20 Jan 2026).

4. Evaluation Metrics and Comparative Analysis

RF mandates rigorous measurement of forecast performance using objective metrics:

• Mean Absolute Error (MAE): $$\mathrm{MAE} = \frac{1}{N} \sum_{j=1}^N |y_j - \hat{y}_j|$$
• Mean Squared Error (MSE): $$\mathrm{MSE} = \frac{1}{N} \sum_{j=1}^N (y_j - \hat{y}_j)^2$$
• Weighted Interval Score (WIS): $$\mathrm{WIS}(F, y) = \frac{1}{K+1/2} \left(\frac{1}{2} |y - m| + \sum_{k=1}^K \frac{\alpha_k}{2} \mathrm{IS}_{\alpha_k}(F, y)\right)$$
• 95% Prediction Interval Coverage: $$\mathrm{Cov}_{95\%} = \frac{1}{N} \sum_{j=1}^N \mathbf{1}\{L_j \leq y_j \leq U_j\}$$

In LLM forecasting, the Brier score $B(p, y) = (p - y)^2$ and calibration error (ECE) are standard measures (Li et al., 20 Jan 2026). Comparative RF results reveal:

• ARIMA and GAM excel at 1–2 week horizons (National 1 wk: ARIMA MSE=0.65, WIS=0.35).
• n-sub-epidemic unweighted ensembles (EM 3 UW) outperform all other models at 3–4 week horizons (National 3 wk MAE=0.74, WIS=0.56).
• SLR and Prophet consistently underperform.
• Regional effects are significant, with Midwest and West favoring ensemble methods for longer horizons, while Northeast exhibits highest errors, requiring wider PIs (Alhassan et al., 30 Nov 2025).

For LLMs, the "SI–TI gap" quantifies leakage, revealing SI forecasts systematically outperform genuine ignorance (TI) by 52% in Brier score (e.g., SI Brier≈0.13 vs TI≈0.26 under Direct+Cutoff). Reasoning-optimized models demonstrate larger SI–TI gaps despite superior overall TI performance.

5. Limitations and Leakage in LLM-Based RF

The integrity of RF depends critically on the model’s ability to avoid knowledge leakage. In state-of-the-art LLMs, SI (prompting the model to suppress pre-cutoff knowledge) does not reliably emulate TI (genuine ignorance):

• Cutoff instructions close only ≈48% of the original no-prompt gap (Direct+Cutoff $\Delta_{1'}$ ≈ 0.13 Brier units).
• CoT reasoning fails to suppress prior knowledge, even in traces that avoid explicit post-cutoff references.
• RL-trained reasoning-optimized models leak more, crafting post-hoc rationalizations that steer toward the memorized answer, hiding leakage behind coherent traces (Li et al., 20 Jan 2026).
• Domain effects: cutoff prompts are more effective in highly structured domains (91% gap closure in Business), but poor in high-salience domains (geopolitics, only 33%).

Surface compliance in chain-of-thought traces is not a reliable leakage indicator; genuine temporal separation is necessary for valid evaluation.

6. Best Practices and Recommendations for RF Implementation

Empirical studies recommend:

• Restricting RF benchmarks to fully post-cutoff events for LLMs, or adopting continuously-updating datasets that preclude outcome leakage (e.g., ForecastBench).
• Using performance gaps (SI vs TI) rather than trace audits to detect knowledge leakage.
• Employing unweighted ensembles for epidemic forecasting at longer horizons due to consistent calibration properties and robustness.
• Iterative retrospective evaluation to identify systematic biases, recalibrate models, and tailor regional deployment.

Parameter-level unlearning should be coupled with rigorous verification protocols; prompt-based memory suppression in LLMs is insufficient (Li et al., 20 Jan 2026).

7. Impact and Future Dimensions

RF remains indispensable for methodologically rigorous model evaluation in forecasting. In epidemiology, RF enables public health agencies to calibrate forecasts, enhance intervention planning, and select context-appropriate models based on region and forecast horizon (Alhassan et al., 30 Nov 2025). In computational forecasting with LLMs, RF highlights the necessity of temporal separation and reveals qualitative and quantitative limitations in simulated ignorance approaches. The ongoing constriction of clean evaluation datasets, due to increasingly recent LLM knowledge cutoffs, underscores the need for new benchmark strategies and real-time, continuously updated evaluation infrastructures. A plausible implication is that retrospective benchmarking of LLMs will become increasingly challenging as generative models integrate wider-ranging, near-real-time corpora, demanding methodological innovation in forecast evaluation and model unlearning.