# Bayesian optimization for non-Gaussian noise

A black box function $f: \mathbb{R}^n \rightarrow \mathbb{R}$, which is evaluated pointwise subject to Gaussian noise, i.e., $f(x) + \mathcal{N}(\mu(x),\sigma(x)^2)$, can be minimized using Bayesian optimization where a Gaussian Process is used as a noisy function model.

How can Bayesian optimization be used for functions subject to non-Gaussian noise, e.g., skewed distributions?

Are there any implementations that support this setting?

• Just a comment: People don't usually use Gaussian process (& normal distributions for all other problems) because of believing that all the things are normally distributed, but because it makes computations easier. – Tim Jan 21 at 15:04

There are Gaussian process models with non-Gaussian likelihood: The prior distribution on the function $$f$$ is still a Gaussian process but the noise term is not Gaussian anymore, i.e. the likelihood $$p(y | f)$$ is not assumed to be Gaussian anymore. As a consequence the analytical results are lost and drawing inference now requires approximation methods such as MCMC or Laplace approximation.
The only articles that I can link you to right now (because I bookmarked them at some point) are on the Student's $$t$$ distribution: