In the Metropolis Hastings version of Markov Chain Monte Carlo (MCMC) I believe that the proposal chain can have self-loops, and those self loops will always lead to transitions that will be accepted. Is that right? If so, are there any recommendations about how to set the self-loops? My feeling is that given that self-loops are always accepted, their impact on convergence is huge.
More details: $$ a(i,j) = \frac{\pi_j T_{ji}}{\pi_i T_{ij}} $$
If we have a self loop in the proposal chain, $\pi_i=\pi_j$ and $T_{ji} = T_{ij}$ therefore, $a(i,j)=1$ so the self-loops will be always accepted. Is that right?
Self-loops are accepted, but are exceedingly unlikely if the proposal distribution is chosen well. For a loop to have non-negligible effects on asymptotic convergence, it would require getting the same samples from the proposal distribution at the same points in the loop infinitely often. This is obviously not the case for most standard choices of proposal distribution.