# Distribution regression

‘Distribution regression’ refers to the situation where a response $$Y$$ depends on a covariate $$P$$ where $$P$$ is a probability distribution. The model is $$Y=f(P)+\mu$$ where $$f$$ is an unknown regression function and $$\mu$$ is a random error. Typically, we do not observe $$P$$ directly, but rather, we observe a sample from $$P .$$

## References

Bachoc, F., F. Gamboa, J. Loubes, and N. Venet. 2018. “A Gaussian Process Regression Model for Distribution Inputs.” IEEE Transactions on Information Theory 64 (10): 6620–37. https://doi.org/10.1109/TIT.2017.2762322.
Bachoc, Francois, Alexandra Suvorikova, David Ginsbourger, Jean-Michel Loubes, and Vladimir Spokoiny. 2019. “Gaussian Processes with Multidimensional Distribution Inputs via Optimal Transport and Hilbertian Embedding.” April 11, 2019. http://arxiv.org/abs/1805.00753.
Poczos, Barnabas, Aarti Singh, Alessandro Rinaldo, and Larry Wasserman. 2013. “Distribution-Free Distribution Regression.” In Artificial Intelligence and Statistics, 507–15. PMLR. http://proceedings.mlr.press/v31/poczos13a.html.

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