I am trying to understand the difference of the initial transient problem and pseudo-convergence when it comes to MCMC convergence diagnostics.
I found the following explanations for the problems:
The term “initial transient” refers to that initial segment of the simulation that is contaminated by bias introduced by starting the system in some state that is not typical of the long-run behavior of the system.
A Markov chain can appear to have converged to its equilibrium distribution when it has not. This happens when parts of the state space are poorly connected by the Markov chain dynamics: it takes many iterations to get from one part to another. When the time it takes to transition between these parts is much longer than the length of simulated Markov chain, then the Markov chain can appear to have converged but the distribution it appears to have converged to is the equilibrium distribution conditioned on the part in which the chain was started. We call this phenomenon pseudo-convergence.
I have troubles to differentiate between the two problems. Both depend on the starting value of the chain. The initial transient problem is resolved by including a "warm-up" / removing samples influence by the initializing value. Pseudo-convergence has to be detected by diagnostics such as multi-chain approaches or investigating long-term behavior and then has to be addressed differently.
Can the two be clearly differentiated? Because to me, they seem to be connected where pseudo-convergence is a "extension" of the initial transient problem. If for instance, the chain is initialized in a area that is not typical for long term behavior and the state spaces are poorly connected in my mind that would lead to pseudo-convergence.
Do you think these two things are connected? If not, could you explain the difference?
References:
Asmussen, Søren; Glynn, Peter W.; Thorisson, Hermann, Stationarity detection in the initial transient problem, ACM Trans. Model. Comput. Simul. 2, No. 2, 130-157 (1992). ZBL0842.68106.>
Geyer, Charles J., Introduction to Markov chain Monte Carlo, Brooks, Steve (ed.) et al., Handbook of Markov chain Monte Carlo. Boca Raton, FL: CRC Press (ISBN 978-1-4200-7941-8/hbk; 978-1-4200-7942-5/ebook). Chapman & Hall/CRC Handbooks of Modern Statistical Methods, 3-48 (2011). ZBL1229.65014.