Understanding and Addressing the Unbounded “Likelihood” Problem

The joint probability density function, evaluated at the observed data, is commonly used as the likelihood function to compute maximum likelihood estimates. For some models, however, there exist paths in the parameter space along which this densityapproximation likelihood goes to infinity and maximum likelihood estimation breaks down. In applications, all observed data are discrete due to the round-off or grouping error of measurements. The “correct likelihood” based on interval censoring can eliminate the problem of an unbounded likelihood. This paper categorizes the models leading to unbounded likelihoods into three groups and illustrates the density breakdown with specific examples. We also study the effect of the round-off error on estimation, and give a sufficient condition for the joint density to provide the same maximum likelihood estimate as the correct likelihood, as the round-off error goes to zero.