# Probabilistic Hierarchical Forecasting with Deep Poisson Mixtures

arXiv: https://arxiv.org/abs/2110.13179 · HTML: https://mlsanity.com/paper/2110.13179/

## Key facts

- Cited by 16 papers in the map (data through 14 Oct 2026).
- Builds on 15 of its 75 references that are in the map.
- First posted on arXiv 25 Oct 2021.

## Builds on (most cited first, 15 of 15)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [WaveNet: A Generative Model for Raw Audio](https://mlsanity.com/paper/1609.03499/) (2016)
- [Spatio-Temporal Graph Convolutional Networks: A Deep Learning Framework for Traffic Forecasting](https://mlsanity.com/paper/1709.04875/) (2017)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Skillful Precipitation Nowcasting using Deep Generative Models of Radar](https://mlsanity.com/paper/2104.00954/) (2021)
- [A Multi-Horizon Quantile Recurrent Forecaster](https://mlsanity.com/paper/1711.11053/) (2017)
- [MXNet: A Flexible and Efficient Machine Learning Library for Heterogeneous Distributed Systems](https://mlsanity.com/paper/1512.01274/) (2015)
- [Neural basis expansion analysis with exogenous variables: Forecasting electricity prices with NBEATSx](https://mlsanity.com/paper/2104.05522/) (2021)
- [MQTransformer: Multi-Horizon Forecasts with Context Dependent and Feedback-Aware Attention](https://mlsanity.com/paper/2009.14799/) (2020)
- [Simultaneously Reconciled Quantile Forecasting of Hierarchically Related Time Series](https://mlsanity.com/paper/2102.12612/) (2021)
- [Probabilistic forecast reconciliation under the Gaussian framework](https://mlsanity.com/paper/2103.11128/) (2021)
- [Hierarchically Regularized Deep Forecasting](https://mlsanity.com/paper/2106.07630/) (2021)
- [Local scale invariance and robustness of proper scoring rules](https://mlsanity.com/paper/1912.05642/) (2019)
- [Approximate Bayesian Inference in Linear State Space Models for Intermittent Demand Forecasting at Scale](https://mlsanity.com/paper/1709.07638/) (2017)

## Built on by (most cited first, 16 of 16)

- [TimeGPT-1](https://mlsanity.com/paper/2310.03589/) (2023)
- [N-HiTS: Neural Hierarchical Interpolation for Time Series Forecasting](https://mlsanity.com/paper/2201.12886/) (2022)
- [MQRetNN: Multi-Horizon Time Series Forecasting with Retrieval Augmentation](https://mlsanity.com/paper/2207.10517/) (2022)
- [End-to-End Probabilistic Framework for Learning with Hard Constraints](https://mlsanity.com/paper/2506.07003/) (2025)
- [HierarchicalForecast: A Reference Framework for Hierarchical Forecasting in Python](https://mlsanity.com/paper/2207.03517/) (2022)
- [Scalable Probabilistic Forecasting in Retail with Gradient Boosted Trees: A Practitioner's Approach](https://mlsanity.com/paper/2311.00993/) (2023)
- [spadesuit SPADE spadesuit Split Peak Attention DEcomposition](https://mlsanity.com/paper/2411.05852/) (2024)
- [Dirichlet Proportions Model for Hierarchically Coherent Probabilistic Forecasting](https://mlsanity.com/paper/2204.10414/) (2022)
- [A More Realistic Evaluation of Cross-Frequency Transfer Learning and Foundation Forecasting Models](https://mlsanity.com/paper/2509.19465/) (2025)
- [Temporal Saliency Detection Towards Explainable Transformer-based Timeseries Forecasting](https://mlsanity.com/paper/2212.07771/) (2022)
- [DeepHGNN: Study of Graph Neural Network based Forecasting Methods for Hierarchically Related Multivariate Time Series](https://mlsanity.com/paper/2405.18693/) (2024)
- [TAT: Temporal-Aligned Transformer for Multi-Horizon Peak Demand Forecasting](https://mlsanity.com/paper/2507.10349/) (2025)
- [Forking-Sequences: Statistically and Computationally Efficient Multi-Horizon Forecasting with Reduced Volatility](https://mlsanity.com/paper/2510.04487/) (2025)
- [Hierarchically Coherent Multivariate Mixture Networks](https://mlsanity.com/paper/2305.07089/) (2023)
- [HiPerformer: Hierarchically Permutation-Equivariant Transformer for Time Series Forecasting](https://mlsanity.com/paper/2305.08073/) (2023)
- [Hierarchical Forecast Reconciliation on Networks: A Network Flow Optimization Formulation](https://mlsanity.com/paper/2505.03955/) (2025)

## Abstract (arXiv)

Hierarchical forecasting problems arise when time series have a natural group structure, and predictions at multiple levels of aggregation and disaggregation across the groups are needed. In such problems, it is often desired to satisfy the aggregation constraints in a given hierarchy, referred to as hierarchical coherence in the literature. Maintaining coherence while producing accurate forecasts can be a challenging problem, especially in the case of probabilistic forecasting. We present a novel method capable of accurate and coherent probabilistic forecasts for time series when reliable hierarchical information is present. We call it Deep Poisson Mixture Network (DPMN). It relies on the combination of neural networks and a statistical model for the joint distribution of the hierarchical multivariate time series structure. By construction, the model guarantees hierarchical coherence and provides simple rules for aggregation and disaggregation of the predictive distributions. We perform an extensive empirical evaluation comparing the DPMN to other state-of-the-art methods which produce hierarchically coherent probabilistic forecasts on multiple public datasets. Comparing to existing coherent probabilistic models, we obtain a relative improvement in the overall Continuous Ranked Probability Score (CRPS) of 11.8% on Australian domestic tourism data, and 8.1% on the Favorita grocery sales dataset, where time series are grouped with geographical hierarchies or travel intent hierarchies. For San Francisco Bay Area highway traffic, where the series' hierarchical structure is randomly assigned, and their correlations are less informative, our method does not show significant performance differences over statistical baselines.