# Probabilistic forecast reconciliation under the Gaussian framework

arXiv: https://arxiv.org/abs/2103.11128 · HTML: https://mlsanity.com/paper/2103.11128/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Mar 2021.

## Built on by (most cited first, 13 of 13)

- [Graph-based Time Series Clustering for End-to-End Hierarchical Forecasting](https://mlsanity.com/paper/2305.19183/) (2023)
- [Probabilistic Hierarchical Forecasting with Deep Poisson Mixtures](https://mlsanity.com/paper/2110.13179/) (2021)
- [HierarchicalForecast: A Reference Framework for Hierarchical Forecasting in Python](https://mlsanity.com/paper/2207.03517/) (2022)
- [Conformal Prediction for Hierarchical Data](https://mlsanity.com/paper/2411.13479/) (2024)
- [When Rigidity Hurts: Soft Consistency Regularization for Probabilistic Hierarchical Time Series Forecasting](https://mlsanity.com/paper/2206.07940/) (2022)
- [Probabilistic Reconciliation of Count Time Series](https://mlsanity.com/paper/2207.09322/) (2022)
- [Large Scale Hierarchical Industrial Demand Time-Series Forecasting incorporating Sparsity](https://mlsanity.com/paper/2407.02657/) (2024)
- [Efficient probabilistic reconciliation of forecasts for real-valued and count time series](https://mlsanity.com/paper/2210.02286/) (2022)
- [Hierarchically Coherent Multivariate Mixture Networks](https://mlsanity.com/paper/2305.07089/) (2023)
- [When Rigidity Hurts: Soft Consistency Regularization for Probabilistic Hierarchical Time Series Forecasting](https://mlsanity.com/paper/2310.11569/) (2023)
- [CoRe: Coherency Regularization for Hierarchical Time Series](https://mlsanity.com/paper/2502.15983/) (2025)
- [Hierarchical Industrial Demand Forecasting with Temporal and Uncertainty Explanations](https://mlsanity.com/paper/2603.06555/) (2026)
- [FoReco and FoRecoML: A Unified Toolbox for Forecast Reconciliation in R](https://mlsanity.com/paper/2604.27696/) (2026)

## Abstract (arXiv)

Forecast reconciliation of multivariate time series is the process of mapping a set of incoherent forecasts into coherent forecasts to satisfy a given set of linear constraints. Commonly used projection matrix based approaches for point forecast reconciliation are OLS (ordinary least squares), WLS (weighted least squares), and MinT (minimum trace). Even though point forecast reconciliation is a well-established field of research, the literature on generating probabilistic forecasts subject to linear constraints is somewhat limited. Available methods follow a two-step procedure. Firstly, it draws future sample paths from the univariate models fitted to each series in the collection (which are incoherent). Secondly, it uses a projection matrix based approach or empirical copula based reordering approach to account for contemporaneous correlations and linear constraints. The projection matrices are estimated either by optimizing a scoring rule such as energy or variogram score, or simply using a projection matrix derived for point forecast reconciliation. This paper proves that (a) if the incoherent predictive distribution is Gaussian then MinT minimizes the logarithmic scoring rule; and (b) the logarithmic score of MinT for each marginal predictive density is smaller than that of OLS. We show these theoretical results using a set of simulation studies. We also evaluate them using the Australian domestic tourism data set.