The product you described is not accurate. You are on the right track with thinking about the location of the alpha hydrogen. The tertiary alpha hydrogen is lost, making a nucleophile that attacks the ketone carbon on another molecule. This is kind of a funny reaction because the intial Beta-Ketone-Alcohol can't dehydrate to give a double bond in the expected location. Normally the double bond always forms in a paticular location because it's energetically favorable.
Do you see why the alcohol normally dehydrates in a paticular way, and why it can't do that in this case?