State-Dependent Mixing

Green (1995) proposed a more general form of the algorithm using state-dependent mixing. There are a finite or infinite set of proposal kernels , , which are permitted to be substochastic. The requirements on the proposal kernels are

• is known for all i.

• 

• For all i

   

  is known, and it is possible to evaluate for all x and y. A different symmetric measure may be used for each i.

• for each x and i, it is possible to simulate realizations from the distribution having the normalized proposal distribution

   

1. Choose a proposal kernel with probability . With probability , skip the remaining steps and stay at x
2. Simulate defined by (1.10).
3. Evaluate Green's ratio

    

   where is defined by (1.9).

4. Accept y with probability .

All of this is just the MHG update described in preceding section combined with the idea of state-dependent mixing.