What are local maxima?
Ideally, the estimation algorithm converges on the globally best solution–the one set of parameter values, out of all possible values, with the largest loglikelihood. Sometimes, though, an estimation algorithm may instead converge on a local maximum solution. A local maximum solution is the best solution in a neighborhood of the parameter space, but not the global maximum. The problem is like climbing a mountain in the dark. By proceeding constantly uphill, always taking the steepest slope, you will reach the top of whatever peak you are already on. However, the highest peak may actually be across a valley; to reach it, you would need to first go downhill, and then uphill again. Finding a global maximum can be difficult for most estimation algorithms, because their strategy is to move “uphill” at all times. Local maxima are related to the complexity of the model; they become more common as the number of latent classes increases. For example, with say eight dichotomous items and only t
