When the state x is a vector 
, a widely used
form of Metropolis-Hastings updating involves only one variable at a time.
Suppose 
is a product measure 
.
To update the ith variable, we need a proposal density 
with respect to 
. The update proceeds as follows
. (Note that y has the dimension
of 
not x.) Let 
denote the state with replaced by y
with probability 
.
Otherwise stay at x.