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 density-approximation likelihood goes to infinity and maximum likelihood estimation breaks down. In all applications, however, observed data are really 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 article categorizes the models leading to unbounded likelihoods into three groups and illustrates the density-approximation breakdown with specific examples. Although it is usually possible to infer how given data were rounded, when this is not possible, one must choose the width for interval censoring, so we study the effect of the round-off on estimation. We also give sufficient conditions for the joint density to provide the same maximum likelihood estimate as the correct likelihood, as the round-off error goes to zero.
This is an Accepted Manuscript of an article published by Taylor & Francis as Liu, Shiyao, Huaiqing Wu, and William Q. Meeker. "Understanding and addressing the unbounded “likelihood” problem." The American Statistician 69, no. 3 (2015): 191-200. DOI: 10.1080/00031305.2014.1003968. Posted with permission.