To a likelihood person the problem of edge effects is a missing data problem. The process is observed in a window, but exists in a larger region. The process outside the window is missing data. The way to handle this is to use the standard procedures for likelihood inference with missing data. One might jump to the conclusion that this means using the EM algorithm, but that would be wrong. The EM algorithm is so slow that when combined with Monte Carlo, making MCEM, it is terribly inefficient. Much better methods exist.