Estimation for General Birth-Death Processes

• Activity Number: 469
• Type: Contributed
• Date/Time: Wednesday, August 1, 2012 : 8:30 AM to 10:20 AM
• Sponsor: Biometrics Section
• Author(s): Forrest Crawford*+ and Vladimir N Minin and Marc A Suchard
• Companies: and University of Washington and University of California at Los Angeles
• Keywords: birth-death process ; EM algorithm ; continuous-time Markov chain ; maximum likelihood estimation ; MM algorithm ; microsatellite evolution

Abstract

Birth-death processes (BDPs) are continuous-time Markov chains that track the number of "particles" in a system over time. While widely used in population biology, genetics and ecology, statistical inference of particle birth and death rates remains limited to restrictive linear BDPs in which per-particle birth and death rates are constant. Researchers often observe the number of particles at discrete times, necessitating data augmentation procedures such as the EM algorithm to find maximum likelihood estimates. The E-step in the EM algorithm is almost never available in closed form; previous work has resorted to approximation or simulation. Remarkably, we show that the E-step can be expressed as a convolution of transition probabilities for any general BDP with arbitrary rates. We accomplish this using a continued fraction representation of the Laplace transforms of the transition probabilities that outperforms competing methods. We derive fast EM algorithms for general BDPs with various rate models, including generalized linear models. We validate our approach using synthetic data and apply our methods to estimation of mutation parameters in microsatellite evolution.