‘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.