i stumbled upon the following formula that describes making predictions based on the MAP estimate. i understand that we "plug in" in the MAP estimate in the predictive posterior, but i do not understand how we get from the second line to the third line, and why we lose the integral and the second term. could anyone help me understand this? Many thanks in advance
Addressing only the question you asked, it's because once you plug in the MAP estimate the first probability no longer depends on theta. Since it doesn't (and the integral is in terms of theta) you can pull it out of the integral and the resulting integral goes to 1, leaving you with the third line.