The Metropolis-Hastings-Green Update

All of the schemes described up to this point used to have separate theory. All are now special cases of the Metropolis-Hastings-Green (MHG) algorithm (Green, 1995), which is essentially Metropolis-Hastings with measures rather than densities. Green was not the first to present an algorithm of this type. Multigrid methods in statistical physics are special cases, as are the methods for point processes presented by Geyer and Møller (1994), but Green gave the first general formulation.

The MHG update is best understood by comparison with the Metropolis-Hastings update.

• The unnormalized density h is replaced by an unnormalized measure on the state space .
• The proposal density q(x, y) is now replaced by a proposal kernel Q(x, A).
• We also need a symmetric measure on to play the role played by in the ordinary Metropolis-Hastings algorithm.

The measure must dominate so that there is a Radon-Nikodym derivative

which replaces h(x) q(x, y) in the ordinary Metropolis-Hastings update. Then the Hastings ratio becomes `Green's ratio'

 

The update is

1. Simulate .
2. Evaluate Green's ratio (1.8)
3. Accept y with probability .

• State-Dependent Mixing