It is enough to verify that elementary MHG update is reversible. Define
the probability of accepting a proposed step from x to y and
the probability of staying at x when the proposal 
is used. Then
is the transition kernel of the Metropolis-Hastings-Green update, and
The first term on the right hand side is obviously symmetric under interchange of u and v. The second term
is symmetric under interchange of u and v because of the
symmetry of 
and because of the MHG update has been defined to make
hold for all i, x, and y. To see this suppose 
in
(1.11). Then
and (1.12) holds.
(1.12) also holds when 
by the same argument.
