What to Do When Your Hessian is Not Invertible

What should a researcher do when statistical analysis software terminates before completion with a message that the Hessian is not invertible? The standard textbook advice is to respecify the model, but this is another way of saying that the researcher should change the question being asked. Obviously, however, computer programs should not be in the business of deciding what questions are worthy of study. Although noninvertable Hessians are sometimes signals of poorly posed questions, nonsensical models, or inappropriate estimators, they also frequently occur when information about the quantities of interest exists in the data through the likelihood function. The authors explain the problem in some detail and lay out two preliminary proposals for ways of dealing with noninvertable Hessians without changing the question asked.