From Wikipedia, the free encyclopedia

[edit] Stationary Distribution

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)

[edit] Proved vs. Proven

Isn't "proven" an adjective, with the more accepted past participle of "prove" being "proved"? Quantling (talk) 11:56, 12 October 2007

I made the change to "proved". Quantling (talk) 17:00, 14 May 2008 (UTC)