Talk:Gibbs sampling
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I'm not an expert, but the phrase 'the stationary distribution' suggests uniqueness to me, but my intuition suggests there could be cases where it isn't unique. For example p(x,y)=0 unless |x-y|<1/2 and 1<|x-y|<2, in which case p(x,y)=1.
- I think there is a typo in your example. If |x-y|<1/2 then it is never the case that |x-y|>1...
- Anyway you are correct that for some distributions with inaccessible regions of state space, Gibbs sampling does not give a unique stationary distribution. In these cases Gibbs sampling by itself isn't a valid MCMC method. Here's a toy example of discrete x and y taking on values 0 and 1.
- p(x=0,y=0)=0
- p(x=0,y=1)=0.5
- p(x=1,y=0)=0.5
- p(x=1,y=1)=0
- Gibbs sampling can never move out of (0,1) or (1,0). However, it does leave the distribution of interest invariant. So Gibbs sampling is always a valid MCMC operator that can be mixed with another operator that ensures ergodicity. Someone should add a lucid description of these issues at some point... 128.40.213.241 13:54, 6 September 2005 (UTC)
