# Variational Learning on Aggregate Outputs with Gaussian Processes

arXiv: https://arxiv.org/abs/1805.08463 · HTML: https://mlsanity.com/paper/1805.08463/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 May 2018.

## Built on by (most cited first, 14 of 14)

- [Deconditional Downscaling with Gaussian Processes](https://mlsanity.com/paper/2105.12909/) (2021)
- [Using Distance Correlation for Efficient Bayesian Optimization](https://mlsanity.com/paper/2102.08993/) (2021)
- [SARN: Structurally-Aware Recurrent Network for Spatio-Temporal Disaggregation](https://mlsanity.com/paper/2306.07292/) (2023)
- [Indirect Query Bayesian Optimization with Integrated Feedback](https://mlsanity.com/paper/2412.13559/) (2024)
- [Learning from Summarized Data: Gaussian Process Regression with Sample Quasi-Likelihood](https://mlsanity.com/paper/2412.17455/) (2024)
- [Conformalized Gaussian processes for online uncertainty quantification over graphs](https://mlsanity.com/paper/2510.06181/) (2025)
- [CLAM: Causal Spatial Disaggregation to Infer Local Effects From Coarse Data](https://mlsanity.com/paper/2608.08064/) (2026)
- [Aggregate Learning for Mixed Frequency Data](https://mlsanity.com/paper/2105.09579/) (2021)
- [Entropic Herding](https://mlsanity.com/paper/2112.11616/) (2021)
- [AODisaggregation: toward global aerosol vertical profiles](https://mlsanity.com/paper/2205.04296/) (2022)
- [Aggregated Multi-output Gaussian Processes with Knowledge Transfer Across Domains](https://mlsanity.com/paper/2206.12141/) (2022)
- [Active Learning for Regression with Aggregated Outputs](https://mlsanity.com/paper/2210.01329/) (2022)
- [Puffin: Probabilistic Learning of Spatial Detail From Coarse Observations](https://mlsanity.com/paper/2610.11914/) (2026)
- [Interpretable Graph Partitioning for Spatial Prediction](https://mlsanity.com/paper/2610.16063/) (2026)

## Abstract (arXiv)

While a typical supervised learning framework assumes that the inputs and the outputs are measured at the same levels of granularity, many applications, including global mapping of disease, only have access to outputs at a much coarser level than that of the inputs. Aggregation of outputs makes generalization to new inputs much more difficult. We consider an approach to this problem based on variational learning with a model of output aggregation and Gaussian processes, where aggregation leads to intractability of the standard evidence lower bounds. We propose new bounds and tractable approximations, leading to improved prediction accuracy and scalability to large datasets, while explicitly taking uncertainty into account. We develop a framework which extends to several types of likelihoods, including the Poisson model for aggregated count data. We apply our framework to a challenging and important problem, the fine-scale spatial modelling of malaria incidence, with over 1 million observations.