A question about the multistart heuristic and pseudo convergence I'm teaching myself MCMC methods and I encountered this passage in a book that I am not able to make head or tails of:

The phenomenon of pseudo-convergence has led many people to the idea of comparing 
  multiple runs of the sampler started at different points. If the multiple runs appear to 
  converge to the same distribution, then—according to the multistart heuristic—all is well. 
  But this assumes that you can arrange to have at least one starting point in each part of the 
  state space to which the sampler can pseudo-converge. If you cannot do that—and in the 
  black box situation you never can—then the multistart heuristic is worse than useless: it 
  can give you confidence that all is well when in fact your results are completely erroneous.

Can anyone explain this a little better?
 A: It isn't clear to me whether you're asking what pseudo-convergence is, or whether you're asking how multistart can fail to detect that pseudo-convergence is happening, so I'll try to answer both briefly.  
Imagine that you're using MCMC to sample from a distribution where $X$ has a high probability (say 99.9%) of being uniformly distributed between 0 and 1, and a small probability (say 0.1%) of being uniformly distributed between 1,000,000 and 1,000,001.  This is a black box problem, so in constructing the proposal distribution for the MCMC sampler you might do something naive like considering jumps that are N(0,1) from the current point.  Under these circumstances, if you start at a point in [0,1], then it's virtually impossible that your MCMC sequence will ever reach the isolated interval [1,000,000, 1,000,001].  Your MCMC sampler will then converge very nicely to a U(0,1) distribution.  Conventional convergenece tests will look great, but you'll have converged to an incorrect distribution.  
You might argue that missing that other 0.1% of the distribution really doesn't matter.  Depending on what aspect of the distribution you're interested in, this could be a very serious problem.  For example, if you want to estimate $E[X]$, you'll get 0.5 as your estimate, when it should be around 1,000!
This is an example of false or pseudo convergence.  
Now, suppose you pick a bunch of starting points and run the MCMC sampler from each of those starting points.  As long as each of these starting points is close to [0,1], each MCMC run will eventually converge to a U(0,1) distribution, and you'll be led to believe that $X$ really does have a U(0,1) distribution.  That's the potential danger of relying on the multistart heuristic.
The example that I've used here is admittedly extreme.  However, pseudo convergence is often reported in practice, particularly on problems in which we're sampling in a very high dimensional space and there are lots of isolated regions with reasonably high probability.   
