How does one know when burn-in doesn't need to be discarded from an MCMC simulation? I'm reading a paper about Bayesian model calibration (https://cfwebprod.sandia.gov/cfdocs/CompResearch/docs/McFCalib0307.pdf). The authors fitted a Bayesian Gaussian process model and sampled from the posterior distribution of the model parameters using a Metropolis algorithm. I'm all good with this, but the following jumped out at me (p. 30)

For each case, 25,000 random samples were generated for the calibration inputs,and no burn-in samples were needed.

Wow! No burn-in required? How would one know a priori that an MCMC algorithm requires zero burn-in?
 A: You need burn-in if you initialize the algorithm in a region of the state-space where the probability density is very low. During the burn-in period, the chain will converge from there to the "typical set", i.e. the region in state-space where probability density is high enough that proposals in this region are likely to be accepted at some point within the finite length of the chain. After burn-in, the chain will only explore this typical set and often won't return to the neighborhood where you initialize it (because the density there is so low that the probability of moving there within the length of the chain is nearly 0). 
In the limit of infinitely long chains, burn-in wouldn't be necessary, since all points in state-space have non-zero possibility of being successfully proposed in the limit of infinitely many proposals. Burn-in therefore is a solution for a problem caused by having finite chains. Without burn-in, the region around the initial point would be over-represented among the resulting samples.
If you are able to pick your initial state intelligently, so that you know for sure that it will be within the typical set, then burn-in may not be necessary. Indeed, this is what the authors say in that paper:

However, convergence is not found to be an issue for the calculations
  done here, given that an appropriate starting value is used (a good
  choice is a value close to the “best” of the true simulator runs).

For instance, the mode of the distribution is usually in the typical set, so if you initialize there you probably don't need to allow a burn-in period. You might know (roughly) where the mode is by construction or domain knowledge, or because you were able to find the global maximum through exact or numerical methods beforehand. 
