Blockwise Metropolis-Hastings Algorithm

In the blockwise Metropolis-Hastings algorithm the candidate density for the block of parameters Oj given all the other parameters 0-j and the data y must dominate the true conditional density in the tails. That is... 

When the Metropolis-Hastings algorithm is run blockwise and the parameters in different blocks are highly correlated, the candidate for a block won t be very far from the current value for that block. Because of this, the chain will move slowly around the parameter space very slowly. We observed this in Figure 6.15 and in the traceplots in Figure 6.16. The traceplots of the parameters of a blockwise Metropolis-Hastings chain look much more like those for a random-walk chain than those for an independent chain. 

Gibbs sampling is the special case of the blockwise Metropolis-Hastings algorithm where the candidate distribution for each block of parameters given all other parameters not in that block is the correct full conditional density for that block. Thus all candidates will be accepted. 

The Gibbs sampling algorithm is a special case of blockwise Metropolis-Hastings where the candidate density for each block of parameters is its correct conditional distribution given all other parameters not in its block and the observed data. The acceptance probability is always 1, so every candidate is accepted. 

---