STA 410/2101, Fall 2015, Assignment 3 Discussion As the initial parameter values for Gibbs sampling, I used estimates based on just the beetles with known species, with an adjustment to avoid values for rho near 0 or 1. This resulted in the Gibbs sampling converging pretty much immediately, as judged by the trace plots, though I discarded the first 20 iterations as burn-in just in case. The resulting estimates from the remaining 480 iterations show posterior means that are mostly similar to the maximum likelihood estimates from Assignment 2 (though those were found without fixing alpha). The trace plots show some substantial dependencies along the chain for some parameters (such as the rho parameter plotted in black), and for some species indicators (such as that for beetle 8). Because of this I increased the length of the chain to 500 iterations, after having initially run for only 200. I also tried using initial parameter values found from averages over all beetles (the same initial values for all species). With these initial values, the chain made a large move to a different place after about 40 iterations, so it obviously had not converged before then. For the remaining 460 iterations, it appeared as if it was sampling correctly, but the distribution of states was in some respects quite different from that found with the first initialization. This was true for a longer run of 5000 iterations as well. It appears that this initialization results in the chain being trapped (from a practical perspective) in a different mode, in which some of the species indicators are swapped, with 5000 iterations not being enough for it to move to the other mode (which probably has higher probability, since it better matches the maximum likelihood estimates, and the estimates based just on the complete data). This is an illustration that Gibbs sampling (and Markov chain Monte Carlo methods in general) can sometimes fail to give the right answer in a run of feasible length.