 Solution

EM likelihood monotonicity makes EM an ascent method, not a global-optimization theorem. If the log-likelihood is bounded above along the iterates, its nondecreasing values converge. That alone does not show that the parameters converge, nor that any limiting parameter is a global maximum-likelihood estimator. Under additional regularity, accumulation points can be stationary; local maxima, other stationary points or flat sets can depend on initialization. Multiple starting values and comparison of the achieved observed likelihoods are therefore useful.

The upper-bound hypothesis is substantive. In a finite mixture model with a freely shrinking normal-component variance, placing one component at an observed value can make the likelihood unbounded while other components cover the remaining observations. In that situation monotonicity provides no finite maximum to converge to. EM may also improve very slowly, and a small change in likelihood alone is not proof of an accurate global solution. The guarantee in part (b) is nondecrease, not strict increase at every iteration.