# How do you use the predictive distribution with noise in Bayesian Optimization?

I have been reading a paper on Bayesian Optimization, and I was reading the section on adding Gaussian noise to your Gaussian process. The article is:

• Brochu, Cora and de Freitas (2010). A Tutorial on Bayesian Optimization of Expensive Cost Functions, with Application to Active User Modeling and Hierarchical Reinforcement Learning. (pdf, specifically: pages 18-19)

The article explains that noise can be modeled with a normal distribution such as $y_i = f(x_i) + \epsilon_i,$ where noise is $\epsilon \sim N (0, \sigma^2).$ Then when calculating the prediction the functions, include the distributions instead of scalars, so how do I input that into the acquisition function (such as the probability of improvement function)?

The functions are on the bottom of page 18 of the link. The mean and variance functions.