The "correct" way to understand this phenomenon is via regression, in which case it boils down to the simple fact that the regression lines E(θ|X) and E(X|θ) ar

The "correct" way to understand this phenomenon is via regression, in which case it boils down to the simple fact that the regression lines E(θ|X) and E(X|θ) are different. (This "Galtonian perspective" is the subject of one of my favorite papers of all time, http://projecteuclid.org/euclid.ss/euclid.ss/1177012274 http://projecteuclid.org/euclid.ss/euclid.ss/1177012274). This is also the only intuitive explanation I am aware of which explains why Stein's phenomenon only occurs in dimensions three and higher.