Bayesian Optimization Process
1 
Use a surrogate model to form a prior distribution of surrogate function that approximate the given blackbox objective function.  Unlike the objective function, surrogate models are cheap to evaluate and allow for the incorporation of prior qualitative information about the objective function (e.g., smoothness). One commonly used surrogate model is Gaussian processes.  
2 
Collect an initial set of data by evaluating the objective function at some number of initial data points.  
3 
Update the posterior distribution using all available data, fitting the surrogate model to the data.  
4 
Sample the objective function at a value determined by the acquisition function to obtain a new data point.  Acquisition functions use the posterior distribution to strategically guide the next sample selection to areas most likely to improve the current estimate. The functions balance two competing goals:


5 
Repeat from (3) until the maximum number of iterations is reached.  The maximum number of iterations is defined by how many evaluations of the expensive blackbox objective function is budgeted. 
This procedure is designed to wellrepresent the blackbox objective function efficiently (i.e., with minimal costly blackbox evaluations) in the regions in which optimal solutions are most likely to exist.