title: bayesianoptimization in machine learning
body: <p>Thanks for reading this. I am currently studying bayesoptimization problem and follow the tutorial. Please see the attachment.<a href="http://haikufactory.com/files/bayopt.pdf" rel="nofollow noreferrer">bayesian optimization tutorial</a></p>
  
  <p>In page 11, about the acquisition function. Before I raise my question I need state my understanding about bayesian optimization to see if there is anything wrong.</p>
  
  <p>First we need take some training points and assume them as multivariable gaussian ditribution. Then we need use acquisiont function to find the next point we want to sample. So for example we use x1....x(t) as training point then we need use acquisition function to find x(t+1) and sample it. Then we'll assume x1....x(t),x(t+1) as multivariable gaussian ditribution and then use acquisition function to find x(t+2) to sample so on and so forth.
    In page 11, seems we need find the x that max the probability of improvement. f(x+) is from the sample training point(x1...xt) and easy to get. But how to get u(x) and that variance here?  I don't know what is the x in the eqaution. It should be x(t+1) but the paper doesn't say that. And if it is indeed x(t+1), then how could I get its u(x(t+1))? You may say use equation at the bottom page 8, but we can use that equation on condition that we have found the the x(t+1) and put it into multivariable gaussian distribution. Now we don't know what is the next point x(t+1) so I have no way to calculate, in my opinion.
    I know this is a tough question. Thanks for answering!!</p>