# Bayesian Deconditional Kernel Mean Embeddings

arXiv: https://arxiv.org/abs/1906.00199 · HTML: https://mlsanity.com/paper/1906.00199/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 1 Jun 2019.

## Built on by (most cited first, 6 of 6)

- [Deconditional Downscaling with Gaussian Processes](https://mlsanity.com/paper/2105.12909/) (2021)
- [Fast Instrument Learning with Faster Rates](https://mlsanity.com/paper/2205.10772/) (2022)
- [An Overview of Causal Inference using Kernel Embeddings](https://mlsanity.com/paper/2410.22754/) (2024)
- [Non-Linear Spectral Dimensionality Reduction Under Uncertainty](https://mlsanity.com/paper/2202.04678/) (2022)
- [Dual Instrumental Method for Confounded Kernelized Bandits](https://mlsanity.com/paper/2209.03224/) (2022)
- [Returning The Favour: When Regression Benefits From Probabilistic Causal Knowledge](https://mlsanity.com/paper/2301.11214/) (2023)

## Abstract (arXiv)

Conditional kernel mean embeddings form an attractive nonparametric framework for representing conditional means of functions, describing the observation processes for many complex models. However, the recovery of the original underlying function of interest whose conditional mean was observed is a challenging inference task. We formalize deconditional kernel mean embeddings as a solution to this inverse problem, and show that it can be naturally viewed as a nonparametric Bayes' rule. Critically, we introduce the notion of task transformed Gaussian processes and establish deconditional kernel means as their posterior predictive mean. This connection provides Bayesian interpretations and uncertainty estimates for deconditional kernel mean embeddings, explains their regularization hyperparameters, and reveals a marginal likelihood for kernel hyperparameter learning. These revelations further enable practical applications such as likelihood-free inference and learning sparse representations for big data.