# Learning-based Probabilistic Load Forecasting with Post-hoc and In-model Uncertainty

arXiv: https://arxiv.org/abs/2607.12730 · HTML: https://mlsanity.com/paper/2607.12730/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 6 of its 38 references that are in the map.
- First posted on arXiv 14 Jul 2026.

## Builds on (most cited first, 6 of 6)

- [Temporal Fusion Transformers for Interpretable Multi-horizon Time Series Forecasting](https://mlsanity.com/paper/1912.09363/) (2019)
- [Review of automated time series forecasting pipelines](https://mlsanity.com/paper/2202.01712/) (2022)
- [Probabilistic electric load forecasting through Bayesian Mixture Density Networks](https://mlsanity.com/paper/2012.14389/) (2020)
- [Energy Forecasting in Smart Grid Systems: A Review of the State-of-the-art Techniques](https://mlsanity.com/paper/2011.12598/) (2020)
- [Lightweight Authenticated Task Offloading in 6G-Cloud Vehicular Twin Networks](https://mlsanity.com/paper/2502.03403/) (2025)
- [A Lightweight DL Model for Smart Grid Power Forecasting with Feature and Resolution Mismatch](https://mlsanity.com/paper/2510.16911/) (2025)

## Abstract (arXiv)

Smart-building load forecasters are often trained offline on dense, multivariate, high-frequency data, but deployment may provide only hourly, feature-limited inputs. Missing features must then be reconstructed, and their errors can propagate through the model. If this input uncertainty is not reflected, prediction intervals may become miscalibrated, affecting demand-response scheduling. Our work examines where uncertainty should be placed once inference inputs are reconstructed. We develop a unified one-day-ahead probabilistic forecasting framework that aligns temporal resolution, reconstructs the unavailable inputs, and derives causal features, and we compare a modular post-hoc residual-quantile scheme with an integrated in-model quantile-learning scheme. The comparison uses three mid-scale Deep Learning (DL) backbones: recurrent, hybrid recurrent, and attention-based Temporal Fusion Transformer (TFT) models, under identical inputs, forecasting horizon, preprocessing rules, and training budgets. Results show that uncertainty placement is backbone-dependent. Integrated quantile learning is most reliable with the TFT, yielding 2.2-3.6% MAPE and 28-83W RMSE on the labeled test window, while producing intervals about 5x narrower than the modular intervals at the closest-to-nominal coverage level. Diebold-Mariano tests support the TFT ranking and the mixed behavior of the recurrent backbones. A reconstruction-sensitivity test shows that reconstructed inputs increase the Quantile Score (QS) by 106% while interval width remains nearly unchanged, indicating that the model does not automatically absorb reconstruction-induced uncertainty. Robustness checks against non-DL baselines and seasonal hold-out weeks support this ranking. Our results expose the limits of post-hoc residual quantiles when inference depends on reconstructed inputs.