"Rafa understood that I wanted to manage a club again. The players understand fully why I've left and they were very supportive on Thursday, my last day," Miller told liverpoolfc.tv.
"I've really enjoyed working with Rafa. He's a very good coach and we had great success in the last four years, when you look at what we've achieved.
"Being the first team coach was a fantastic position to be in at such a top club. I worked exceptionally hard for years with numerous clubs to try to get to the top and he gave me that chance.
"I will always be indebted to him."