Understanding and Addressing the Unbounded 'Likelihood' Problem

• Activity Number: 175
• Type: Contributed
• Date/Time: Monday, August 4, 2014 : 10:30 AM to 12:20 PM
• Sponsor: Section on Physical and Engineering Sciences
• Author(s): Shiyao Liu*+ and William Q. Meeker and Huaiqing Wu
• Companies: Genentech and Iowa State University and Iowa State University
• Keywords: Density approximation ; Interval censoring ; Maximum likelihood ; Round-off error ; Unbounded likelihood

Abstract

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 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. We categorized the models leading to unbounded likelihoods into three groups and illustrated the density break-down with specific examples. We also studied the effect of the round-off error on estimation, and gave 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.